Qfl 303 
.G87 
Copy 1 




Class __GLAS02l 

Book Gc^|l_ 

ftpigliti^" 



COPYRIGHT DEPOSIT. 



AN INTRODUCTION 



TO THE 



SUMMATION OF DIFFERENCES 

' OF A FUNCTION 



AN ELEMENTARY EXPOSITION OE THE NATURE 

OF THE ALGEBRAIC PROCESSES REPLACED 

BY THE ABBREVIATIONS OF THE 

INFINITESIMAL CALCULUS 



BY 



B. F. GROAT. 

Assistant Professor of Mathematics and Mechanics in the 
University of Minnesota 



MINNEAPOLIS 

H. W. WILSON, Publisher 

1902 



THE LIBRARY OF 
CONGRESS, 

Two Copies Received 

FEB 10 1903 

Copyriglit Entry 

XLASS CL_ XXo. No. 

COPY B. 



COPYRIGHT, 1902 

BY 

B. F. GiJOAT 



^PREFACE 

The subject-matter for the following pages has its origin 
in various sources well known to the general mathematical 
reader ; yet it is felt that the view of the differential and inte- 
gral calculus presented in the sequel generally does not come to 
be recognized by the student sufficiently early in his course. 
The view referred to is suggested in a number of excellent 
works upon infinitesimal calculus, but the writer's aim is to treat 
the subject of this paper as a chapter in algebra, preparatory and 
supplementary to a course in differential and integral calculus, 
and not as a part of that course. The historical order of develop- 
ment has been followed in general outline, though the exercises 
are set in modern notation. The historical order of development 
of a science is usually (net always; cf. Hist, logarithms) the 
easiest ^to follow ; moreover any other method withholds history 
and often fails to give a suffici?nt account of the science in- 
volved. This stands to reason for the actual development is like- 
ly to be the result of necessity. The reader is referred to several 
excellent articles and books for the history, which, after reading 
this paper, should be intelligible to the beginner. 

The object is not merely to gain the historical view of the 
infinitesimal analysis br.t to prepare the student for the solution 
cf problems in applied mathematics. The processes of difi'er- 
entiation and integration are acquired without much difficulty; 
but to see the integral with facility in a problem in analytic me- 
chanics or physics requires clear notions as to sums and limits 
of sums. Such notions are of much more importance to the 
]:)hysicist and engineer than the more elaborate methods for in- 



tegrating complicated forms : it is the desire to aid the student 
in forming these notions early, together with the writer's need 
of a suitable exercise book for use in his classes, that has been 
the reason for writing this paper. 

The following is suggested as a course in fundamental prin- 
ciples and exercises : . Elementary algebra, including progres- 
sions ; convergence and divergence of series including the 
elementary test theorems ; ^sums of squares and cubes of 
first n integers; undetermined coefficients and decomposition 
of fractions ; exponential and logarithmic series and logarithms ; 
elements of trigonometry; summation of series as in Chapter I, 
this book ; derived functions ; theory of equations, graphs and 
elementary limit theorems as stated in Arts. 7, 8, 9. and Theorem I, 
Art. 14, McMahon and Snyder's Differential Calculus; permu- 
tations, combinations, binomial theorem ; determinants, system 
of linear ecjuations, elimination, Sylvester's Method, discrimi- 
nants; analytic geometry of plane and space; Chapter II, this 
book ; difif erential calculus proper, that is, the rules and formulas 
of differentiation ; Chapters III and IV ; integral calculus fol- 
lowed by more complete courses. 

For a simple demonstration of the logarithmic series the fol- 
lowing is suggested: £>''o* (^-^^e (in-vr)->'; expanding by the ex- 
ponential and binomial theorems and equating the coefficients of 
y in the two expansions we obtain the logarithmic series. Of 
course the limitations of this proof should be noticed. The 
formulas for the sums of squares and cubes may be proved by 
induction. An early introduction to the factor and remainder 
theorems with their application in drawing graphs and locating 
roots of rational functions is advocated. 

The kindness of Mr. H. H. Dalaker in reading proofs and 
verifying examples is acknowledged. 



University ok Minnesota 
Marcl). 1902 



SUMMATION OF DIFFERENCES 

CHAPTER I 
SUMMATION 

1. The Symbol S. The series with which we shall have to 
deal are such that the ;/th term can be expressed as a function 
of 71. Arithmetical and geometrical series are of this kind. 
Thus the ;/th term of the arithmetical series, a-\-{a-\-d) 
-\-{a-\- 2 d)-\- '■•, is (<^ + ;/ — I <^) ; any term may be written 
from this by substituting the corresponding value of ;/. As an 
example the third term of the series \^ {a -\- }, — i d) = {a -^ 2 d\ 
as it should be. In the geometric series a ■\- ar -\- ar'^ -^ •••, the 
//th term is ar''~^, and from this, for example, the fifth term is 
ar'''~^ = ar*, as it should be. Hence we see that the terms of 
an arithmetic progression are of the type (a -\-x— id), and the 
terms of a geometric progression of the type ^r-^~\ where x is 
to have the particular value corresponding to the number of 
the term in question. 

2. The symbol V is employed for the purpose of indicating 
that a sum of terms is to be taken. We define 

-^(a+x^^id) 

to mean the sum of terms of the type (a + .r — i d), 

X ^^■'"' 

to mean the sum of terms of the type ar''~\ and in general 



2 SUMMATION OF DIFFERENCES 

to mean the sum of terms of the type </)(a'). Our definition, 
however, is not complete, since nothing in the notation indicates 
how many terms of a given type are to be inckided in the sum. 
Suppose, for example, we wish to indicate the sum of the first 
five terms of the series 3, 5, 7, 9, ••• ; we should write simply 

V (3 + 2 -r — I ), and have to explain that we are to sum the five 
values found by making x successively equal to i, 2, 3, 4, 5, in 
(3-i-2.r— I). We complete the definition by attaching the 
limits of X in the following manner : 

2(3 + 2.7:^); 

.r=l 

the limits designating the first and last values to be given to x. 
The limits may be omitted in any problem provided they are 
understood. 

3. The student will have no difficulty in verifying the follow- 
ing illustrative examples : 

22(.r+i)=2. 4 + 2-5 + 2. 6 + 2-7 + 2. 8 = ^-^^ - 5 = 60. 

V 2-1 = 2^ + 2'' + 2^ + 2^ + 26 = ^'^^''~ ^^ = ^_liJ. = 124. 



2 — I 



^,r=i+2 + - + 7 = ^7 = 28. 



y .2^ l2 , ^2 , ... I io2_ IQ(tO+ 0(20+ l) ^ 10- II -21 

T" "^"^ 6 6 



%.,{%.'i;.Y-^^^^.. 



5- 11-7 = 385. 



T V x) ^ ir X X A ^ 



^o + g=SoU- 



SUMMATION 



"^ log 'J = log 60. 

i;(-ir'iog- = iog5. 

;S '^ h // 5 6 7 8 9 10 

^^/r(,r) = A/r(;;/) + ',;r(;;/+ l)+ ••• + f{7l - l) + f(/i). 

V sinf — + X jcos f ^^^ xj= - V sin .c/tt + - ^ sin 2 ,i- = 2 sin 2 x 



.2 



4. In the preceding paragraph we were concerned with find- 
ing the vakie of ^(fii^); in the present and following article 

we shall solve the inverse problem, having given a series of 
the form (/)(7//) + (f)(7n + i) + ••• -f (/>(;/) to find the equivalent 

expression ^(f>(x). As an example take the series 

_3 4_ _^ __!_ _ 6 _^ ^__ _ 24 

4-5 5-7 6-9 7- II 25.47 

AVe observe at once that the signs of the even numbered 
terms kre negative ; this is effected in the summation by intro- 
ducing the si£-;i fador (— i )'~\ so that when .c is odd the sign 
is +. We notice further that each numerator is greater by 2 
than the number of the term in which it stands, while the first 
factor of the denominator is 3 greater. The last factor of 
the denominator increases 2 units per term, and is 5 in the 
first term, hence it is 3 + 2 n in the ;/th term ; that is, it exceeds 
twice the number of the term by 3. The ;/th term, therefore, 
expressed as a function of //, may be written 

(;/ + 3)(2;/ + 3) 
and, all together, there are 22 terms. 

. _3 4 , ... 24 ^ v' (- iy-^Cr+2) 

"4-5 5-7 25-47 ,ff (^r+3)(2a'+3) 



4 SUMiMATION OF DIFFERENCES 

5. Let the student verify the following equalities : 

r=10 

r . I 



|4 l5 



+ 



lo r* U- 






--+---+y-2:(-.) 



I \ 

I I 4- -I' 



+ 



=1 



I . I 



4- 

x I + ,1' 2 + ,t 



I + 2,r I 4- 3'^' ::ri I + ^T^i A- 

17 

17 + A- f^,.+ 



+ -r 



in 



log iit^ log 7;^'^ log 111^ ^4 J/ log w 



6. The following identities are obvious, being consequences 
of our definitions and the ordinary rules of algebra : 

+ -+fi;<^(x)}. 



{a) 



Illustration. 



12 4 <; 10 12 

Sz- =Sz- 4-^2' +Vs- +:£02_ 
11 5 7 11 



I. 



y. (i) 



SUMMATION 



5 4 ;» 11 

I LLUSTRATIONS. V - = ^ , V = V -- 

?. Z- ■i{z-\-\)- :, {X^ 2)- 7 X- 






('■) 



Scholium. The braces may be omitted from the second 
members of (<?) and (r), and when there are several summa- 
tions of the same function for different sets of limits, as in (a), 
we may further abbreviate thus : 

t'A-r) - 1 1 + i; + 1 + - +i; l-^ w- 

Scholium. We suppose in this article that none of the 
quantities involved are infinite. 

7. Summation of Series. With the aid of the relations 
treated in the preceding article the summation of many series 
of the kind we are considering can be effected. This can 
always be done if the series is one whose ;/th term can be 
expressed as the difference between two expressions, one of 
which is the same function of (;/ ±/) that the other is of ;/, / 
being an integer. When this is not the case, other artifices 
must be employed, illustrations of which will be given, but the 
device alluded to above is the one of fundamental importance 
and to which most of our attention will be directed. 

EXERCISES 

1. Sum the series //j + //_. + //.. + ••• + //„ + •••, where i/„ is capable 
of being expressed in the form <i>{x)— <^(j; +/)]^=„. 



SOLUllON, 



SUMMATION OF Dli-FEREXCES 



2'^ = 2 



(/>(.V)-</)(.V+/) 



= 5^<^^^')-2<^('^-+/) 



S 2 

L- 1 p-1. 



C/>(A-) 



»! »+;) 






<A(^v) 



2. Find the sum of the (// — /;/+ i) terms from the 7//th to the ;nh, 
l)Oth inclusive, in Kx. i. 



Solution. 



^u, = ^cl>{x)-f^<f>{x+/>) 



XX 



</)(.V+/). 



-m 1 



i^m—p m m «— //— l_l 

m— p— 1 H^ji — 



0(A-+/) 



i:-2: 



-+1-I 



c^Gr) 



2;c^w-2;<^w. 



-+,>-i 



Scholium. There are fewer terms in 2 ~ 2 ^^^^^-^ "^ 2 ^"^-^ 

when 2/ < (;z — ;;/ H- 1). The most important case is when p = i, 
this giving two terms in the result, no matter how many in the given 
sum. The sum to infinity is found by passing to the limit when ;/ =^ cc ; 



thus, 



X^'.. = S*(-)-„k'.S*(-) 



SUMMATION 



3. Find the sum of n terms of 
sum of the series. 

Solution. The //th term is 



" -! h H and also the 

2 2-3 3-4 



But by decomposition of 



n{u ^ t) 



fractions - 



n\n-\-\) n n^\ 

... V ^-i = y i_y JL_ 



I ^ .r A' X + I ;/ + I 



^^ + 



Therefore, 



1-2 2-3 3-4 



4. _^ + _^^-l_ 
1-3 2-4 3.5 



Solution. 



z{z-\- 2) 2^2 s + 2^ 

••• ? — ' — =-ry '-T"^1 



. 1 V u r+lJ 



i+i 



2 ;- H- I ;- + 2 



-2 I 
Hence the sum of r terms from the beginning is 



+ -;- 



r-\- 1 /'+ 2 

II "? 

Passing to the hmit when r = 00, we find that \ -f ... = ^. 



1-3 2-4 



5. y ' ^? 

^ x{x + ^){x-\-6) 



Solution. 



x{x-\-6) 6 



X x-^6 



xix-^^){x-{-6) 6|_jr(>x + 3) (^ 



L^(x + 3) (^ + 3)G^^-h6)J 




SUMMATION OF DIFFERENCES 



Whence V '- = ' ^V ' - V 

x-x 



(.T + 3)(.T+6) 



x+x-x-x 



+ 



4 Jif 1 

I 



I 



4 2-5 3-6 .(;/ + i)(7/ + 4) 
I I 



(;z + 2)(;z + 5) (^ + 3) (^^ 4-6)_ 



The sum to infinity is -^^^ — 
•^ 1080 






Solution. 



^^4-3 



;/(// + i) (/? + 2) 2 
X4-3 



7Z + 3 



^^ + 3 



JC+2 

^ + 3 



7/(7? + I) (7^ + i)(;z + 2) 
r-h3 ^ -'^' + 3 



'Y :^_^r3. = 1 V ^ ^3 _ >y^ r>- -r- 3_ 



2 4_ y __^_Jl_3 y _x-|-_2 ^^ + 3 



+ 2) 



^^ + 3 



2+V '- ^^ 

A'.r(j*:+ I) . (77 + i)(77 



+ 2), 



iL+3 



^ \X X -\- ij (77 + l)(77 



+ 2X 



/ T 



2 +; 



77 -I- -^ 



\2 11 -\- \j {n + l)(77 + 2 



5_i 
4 2 



I 77 + 3 



77 + I ~ (77 + i)(;7 + 2) 
is the sum of n terms. Passing to the limit, we derive the result required. 



SUMMATION 



7. Find the sum o{ n terms of ^ — \- etc. 

3-5 5-7 



Solution'. 



11 -\- 1 



I / // -f I n -\' I 



{2 71 -\- I }{2 n -\- T,) 2 \2 ;? -H I 2 7/ -4- 3 



A(2j'4-i)(2r-h3) 

- + V (- i)'-^ -'^ ' + y ( - O"-' 
.3 T 2V4-I ^ 

.3 T 2/7 + 3 



4^ 2 V + I ^ 2 r -h 



J -H 3. 

2J'+I« 2 77 + 3 



2 I ^(- i)"-' , . _x„ n -\- T 



2 77 + 3_ 



.3 2 2 



8. l + ^+3^...+_JL 
LI i3 [4 l«_± 

Suggestion. 



Ijf + I |.r A" + 



I-2-3 2-3-4 3-4-5 



10. Sum to infinity the series 



3,+ .^, + ^,^ + 



Suggestion 



I- . 2- 2- • 3" 3-.4- 



2 X -'- T 



11. 



x-{x + i)- j:- (x + i)- 

y ' — ='• 

'^ {n 4- 2)n 2 



, ('^ + 2) 

Suggestion 



« + T ;77 + 2 (77 + 2)1 77 



12. Find the sum of any ;- consecutive terms, and also the sum to 
infinity of 



2- — I 4" — I 6- 



lO 



SUMMATION OF DIFFERENCES 



13. -^.1 + ^.1+ -4_..i^^.._p 
1-33 3-53"' 5 • 7 3' 

Suggestion. The ;/th term should reduce to 
i/ I _j. I 



4\2 ^^ 



Alls. - 



2 // + I 3' 



Sum of 71 terms is - 
4 

Sum to infinity is -• 
4 



(2// H- 1)3' 



14. Sum to n terms the series 

cos X + cos {x -\-y) + cos (.r 4- 2 r ) + • • • 

Solution. 2 cos x sin y ee sin {x -\- y) — sin {x —y) . 



.'. 2 cos {x -\- 11 — 1 y) sin -- = sin ix-{-2n— i "- ) — ^ii^ f a: + 2 // — 3 ^ 
whence 

r-\ 



2 sin^ Vcos(jc+;z — ij') = 'Vsinf ^4-2/^— I-- J — Vsin(.v+2;/ — I 



= sin f J*: + 2 r — i '- ) ~ sin { x — -^ 



Hence the sum of n terms is 



sm [ x -\- 2 n — 1-] — sin ( jc — - 



2 sin 



15. V sin {a + zl?) = ? 



16. 



4- 



+ ••• + 



1-3 I • 3 • 5 I -3 -5 • 7 I • 3 . 5---2;z-f I 

Before proceeding with the solution we give a definition quite similar 
h 
to the definition of V cjb(jc), namely : 

a 

Tlxl/{x) = ij/{?n)\{/(m -f- i)i//(/;/H- 2) •■•\p{n). 



SUMMATION 



II 



Solution. 



I -3 • 5---2 n-{-i 



.'.t 



z=\ II 2 Jt: + I 

1=1 



5 •••2 //-I I- 3. 5... 2// + 



' n , n ^ 



z=l U. 2 X — I z=ljl 2 X -\- I 



X 



^=1 n 2 jc -f I ^=1 n 2 jc + I 



x-x 



II 2 JC + I 

x=0 



n 2 X H- 1 

x=0 



I — 



I . 3 . 5 ... 2 ;z+ I. 
Scholium. Solve V, by this method. 



8. Scholium. In solving these examples the sums of the 
{m — Ji+ i) terms from the ;/th to the ;;/th, both inclusive, 
should be determined also. Additional examples will be found 
in C. Smith's Treatise on AUebra. 



CHAPTER II 

LIMITS 

Before beginning the study of this chapter it is necessary to 
know the definition of a Hmit, the meaning of the symbol =, 
and the elementary limit theorems mentioned in the preface. 

9. Theorem. The limit toivard zvJiicJi the vahtcs of the frac- 
tions — j^, -^ — ^ approacJi indefinitely neai% zvhen the value of 
X tan X 

X is taken indefinitely near zero, is unity ; or in notation, 
tan X 



= ^' I 

^^0 tan X 



= I, 



Proof. We know that tan x > x > sin.T.* 

X 

I > > cos X. 

If xj be taken sufficiently near to zero, cosjt* can be made to differ 
from unity by- a quantity as small as you please ; therefore, with greater 

reason, -^ — , which is less than unity but greater than cosx, can, by 
tan X 

taking x small enough, be made to differ as little as you please from 
unity. Hence, the theorem 



tan x^ 

Let the student prove the second part, i.e. 



It should be noticed that we do not say — - — , or 



= I. 



V tan X 



tan X X 

ever does really equal unity, but that unity is the limit toward 

*The student should demonstrate this inequality, referring to some trigonometry 
or geometry if he has forgotten the proof. See Levett and Davison's Plane Trigo- 
nometry, Vol. I, p. 78, Cor. 

t Here we use x to mean the value of x, and hereafter shall frequently do the 
same with reference also to other symbols. 



LIMITS 



13 



which either fraction approaches indefinitely near when x is 
taken indefinitely small. 

Lt tan X Lt ^ 



Cqr. 



tanx 



I. 



10. Theorem. 



sinx 



= I and -4- — 
,^0 sin X 



Let the student give the proof. 



Cor. 



Lt sin.r Lt '^' 



X ^o 



= I. 



sni X 



EXERCISES 



tan ax 
1. 

X 

Solution. 
But 



= ? 



tan ax 



-, which is indeterminate. 



=0 o 



tan ax tan ax tan z . _ 
= a = a , II s = ax 



Lt tan ax _ x.t ^^^ ^^ _ Ix ^^^ ^^ _ Lt tan s 

•^ - O .T ~ X = O ^j^; "~ X = ax ~ 2 = 2 * 



Lt tan ax 



A' = 


Jic 


Lt 


X 


X — 


sin mx 


Lt 


ax 



2 Find 

3. Find 

•^ — o tan jc 

4. (/; — x) cot (^ — x) 



Ans. — 
m 



Ans. a. 



— ? 



Solution. {/? — x) cot (3 — x) 



= o X 00 is indeterminate. 



But (/. - .r) cot (/. - X) ^ — ^4^^. and ^t- ^ ^^^7 ^^^ ^ = i . 



tan (/^ — .t) -^ = ^ tan (<^ — jc) 



Lt 



2^(^ - x) cot (^ - .t) = I. 



14 



SUMMATION OF DIFFERENCES 



5. Find ^-^ ^^ (<r — x) cot (<^ — x), c =^ b. 



SUGGESTION. {c — X^ QQX {b — X) ^ 



6. Find V^(2 — ;»r)tan-- 



{c — x) {b — x) 



{b — x) tan {b — x) 



Ans. 00 . 



(2 — ^) tan 



is indeterminate. 



But (2 — ;r) tan 



Now 



TT 2 — JC 



2 — ji; 



X ^TT ^ fir 7r\ (it 7r\ ( -n -n 
cot - tan tan 

X \2 X) \2 X) \2 X 

(TT TT 

2 X 



tan ( 

2 JC 



I ; hence our answer is 



2 — X' 

TT TT 

2 Jf 



^, , . Lt 2 — J*: Lt 2 JC , , 4 

, that IS ^^2— - = ^^ (-1) = -^. 

Jv — 2 TT TT 

TT 

2 JC 



7. Find the limit, x = o, of j\; tanf axy 

8. Same as Ex. 5, when x = c. 



Afis. 



11. An Application of Principles. Draw a circle of radius a. 
Divide the radius into n (say 8) equal parts, and also the arc of 
the quadrant whose origin is the extremity of the divided radius. 
At each point of division of the radius erect a perpendicular, 
and from each point of division of the quadrant draw a radius. 
Draw a curve through the points of intersection of the perpen- 
diculars and corresponding radii, beginning at the origin. This 
curve is the qitadratrix. Take the origin of the arc for the 
origin, O, of rectangular coordinates, the divided radius being 
the axis of X. Let two points, P and P-^, move uniformly, one 
in the arc of the quadrant, the other along the radius, so that 
both start from the origin at the same instant and arrive, P at 
the extremity of the arc of the quadrant, and P^ at the center, 



LIMITS 15 

C, of the circle, at the same instant. It is clear that the per- 
pendicular from P^ intersects the radius from /^ in a point of 

the curve, and that the vertical intercept is the ordinate y, of 

OP 

that point ; that OP^ = x and = a, the measuring arc to the 

a 

an^le PCO. Also - = — • Let the student show that 
^ X 2a 

/ . '^'^ 

-y = (<7 — .t')tan — 

is the equation of the curve. 

IT fa — X 

, , (a — ,f ) (a — x\ 2 a 2\ a 

Now 1/= -= = 

, irx . (IT 7r,v\ IT IT a — x 

cot — tan tan - 

2 a \2 2 aj 2 \ a 

IT fa — 

2\'~a . 
and 7 r- becomes indefinitely near to unity when x is 

77 a — x\ ■' •' 

'^"2 1- .-J 

taken indefinitely near to a ; therefore 

y 



^2a 

TT 



Hence, if j'q is the ordinate for x = a, 

2 a Diameter of circle 
7o Maximum ordinate 

We have then a mechanical means of determining an approxi- 
mate value of TT, by simply dividing the diameter of the circle by 
the maximum ordinate as found by construction. An approxi- 
mate value of TT (one of the earliest determinations) was found 
in some such manner by Dinostratus, about 400 b.c. The 
curve is frequently called the Quadratrix of Dinostratus, but 
was invented about half a century before the time of his quad- 
rature by Hippias, supposed of Elis. The curve is also em- 
ployed in the trisection {transcendental) of the arc, and, indeed, 
may be used to find any fraction of an arc. An irregular 
draughting curve is of great assistance in the construction. 



l6 SUMMATION OF DIFFERENCES 

Check the construction by plotting two or three points of the 
quadratrix from the equation 

J = (a — x) tan 

2a 

12. Archimedes (born 287 b.c.) of Syracuse, the greatest 
mathematician of antiquity, found that the circumference of a 
circle was " greater than three times the diameter by a fraction 
less than l^ and greater than l-^ of the diameter." His method 
was to inscribe and circumscribe regular polygons of 96 sides 
within and about a circle, and then from actual measurement 
to compare the perimeters with the diameter. Thus if Pj and 
P^ are respectively the perimeters of the inscribed and circum- 
scribed polygons, and R the radius of the circle, he found 

2R ^'^ ^2R ^'1- 

Let the student explain the double inequality and also make 

P P 

an estimate of — ~ and — '- by measurement, after inscribing 

and circumscribing regular polygons of 24 sides in and about 
a circle of 5 or 6 inches radius. The following inequalities 
were established in this way : 

P, 22.8 , ^^ ^ Pi 22.56 

—^= = 3-17 >7r> — h, = — ^= 3.13. 

2R 7.2 ^ ^ 2R 7.2 ^ ^ 



13. Theorem, (i +- 
between 2 and 3. 



= <?, where ^ is a number which lies 



Proof. Let jn be a positive integer. Then 



111 J V in J \ in 



'+n^) ^' + ^+—J-+ 13 






\in 



LIMITS 
by the binomial theorem, and 



m+l I I : 1 II- 



l+-i-- ^1 + 1+^^ ^n+ij_^\ m+lj\ ;;.+ ! 



m + ij ' ' |_2 ■ 1^ 

I \ f m 

I 



VI -\- ij \ ;;/ + I 

4_ ... + i ;;/ + I 



by changing in to in -\- i. Whence it is plain that 



1^1+-^^] >(!+-:) 



in -\- ij \ in J 



f I \"'+i 
since, after two terms, each term of i H is arreater than 

the corresponding term of f i H J , and, moreover, there is 

J \m + l 



one more term in ( i H — . Hence, ivhcn x is increased 

\ in + ij 

by unity, x being any positive integer, f i + - ) increases. 

Again, it is clear that f i H — ) can never be greater than 

the series 

II I ^ I 

This last series is less than 

Whence fi+-y<3- ^ 



;;2 being a positive integer. If we take in to be a very large 

f I ^'" 

positive integer, we see that i H — < 3 however large ;;/ 
may be, since it is always less than a quantity which can never 
be greater than 3. Therefore, since fiH--j increases con- 
tinually when .s does, xr being a positive integer, but at the same 

= 2, we see 



time is always less than 3, and since ( i + - 



5=1 



1 8 SUMMATION OF DIFFERENCES 

that ( I + - ) must approach some limiting vahie which lies 



between 2 and 3. Call this limit ^, and the theorem is proved 
for a positive integer. 

To prove the theorem for a positive fractional value of x. 
Let ^ be a positive fraction, and m and n two consecutive 
integers, such that 

in = 71 -\- I, and in>6 > n. 

Then (:+!)< (: + ■)<(.+! 

that is, (i+l)"-"<(,+^J<(:+ip 

a and /3 being positive proper fractions; hence 

^"i<(-j)'<(-3-(-,7 

mj 

If ^ is made to increase without limit, ;;/ and ;/ consequently 
do the same, 

therefore (i+^j = <^, being a positive fraction, and the 

theorem is proved for positive fractional values of x. 

To prove the theorem for negative values of x. Let x be 
any positive number, and put y = — x ; hence y is numerically 
equal to x, but negative. We have 

■^■)--(-.r-f^r-fer-(-.^ 



I\" / I x^-l 



!'+_,;; -I,' +,—7) !■ + 



.^t) 



LIMITS 19 

Let ;tr = 00 ; then 

i' = -«, (i+.^^)'"'=^'-. and (i+-^-^)=,. 

Hence the Umit, when y approaches negative infinity, of 
(I + i )' is c. 

Therefore i + - J =■ c whether x be positive, negative, 

integral, or fractional. • q.e.d. 

Scholium. For other proofs of this theorem see Todhimter's 
Dijfcrcntial Calcjtlus. 

14. The Value of e. The student may be interested to know 

the value of e. By definition, ^=\-^oo(i+ ) > and this in 

whatever way x is made to approach infinity, whether positive, 
negative, integral, or fractional. Then let vi be a positive 
integer; whence 



I \'" V mj V in) \ VI 

m) [2 







\1L 


— I 




(- 


n 


\-- 


(- 


vi — i\ 


mj 


1 


;« ) 



+ ••• + 

I 7/1 

Let the sum of the first 71 terms of this series be denoted by 
S„, and the remainder after ;i terms by R„ ; that is, 

,_!)...(, _?i^^ f,_i)...^_--i 

K,, = ■. h • • • + 



[u \in_ 

the remaining {in + i — n) terms of S„,^i. 

Hence ^« < ,- + 1—7- + •••+,— 

I r I I 

<-^ I +-+ •••+-^;^ 



20 



SUxMMATION OF DIFFERENCES 
I I 



^n< 



I I 



and 



^d£ _ I 71—1 \ii — I 

n 



1+" =^.+^«- 



Lt 



It is clear that if n be any finite number ^^^ ^ ^ 5,, = i + i 

_l_ — (_..._!_ , which call ?/,, ; that is to say, S,^ = n,^ — p, 

\2_ \n— \ 

where /o is a positive quantity that can be made as small as 

you please by taking vt sufficiently large, n being finite. We 

have then 



11 — \\n 



or 



iin -P< -^n> + l < i^n -P + 



11- I 



Passing to the limit when iii = oc, we have 

I I 

11— I \n — \ 



If n be taken sufficiently large. 



n — \ \n — \ 



can be made as 



small as you please, and therefore, if n is large enough, ii,^ will 



differ from ^ by a quantity smaller than 



n— I ;/ — I 



; that is, 



smaller than a quantity which is as small as you please. Hence 

^ = ?/oo == I + I + 1- + r 4- etc. 

We may now find the value of c to as high a degree of approxi- 
mation as desired. For example. 



-^^tl-^-th^th 



'■^\i 



and 



tb 



< 



^ I 

8|^' 



< 0.000004 



LIMITS 



21 



00 T 

Hence the sum of the first nine terms of i + V - differs from 

e by less than 0.000004, and consequently will give the first six 
figures of e provided the sixth decimal figure of the sum is less 
than 6. For clearness the calculation is arranged as follows : 



1 + 



X 



\i 



< ^- < I + 



» T 



I I 



2.0000000 = 


I + ] 


^li 


= 


2.000000 


.5000000 = 


I - 


-|2 


= 


.500000 


.1666666 < 


I - 


-^ 


< 


.166667 


.0416666 < 


I - 


-L4 


< 


.041667 


.0083333 < 


J _ 


-is 


< 


.008334 


.0013888 < 


I - 


-|6 


< 


.001389 


.0001984 < 


I - 


-Iz 


< 


.000199 


.0000248 < 


I - 


-|8 


< 


.000025 


.0000027 < I 


-l_9; 


I -h8 


8< 


.000004 


.0000002 < I 


^ 1 10 


) 




2.718285 



2.7I828I4 

Hence c lies between 2.7182814 and 2.718285, and conse- 
quently 2.71828 must be the first six figures of e. As a matter 
of fact 2.718281 are the first seven figures; but to prove this 
would require further investigation. By taking in more terms of 
e, and carrying each term sufficiently far out, we may, by reason- 
ing analogous to the above, find the value of c as nearly as 
desired. 

It will transpire that e is the base of the natural system of 
logarithms, an absolutely incommensurable number. It is called 
the Napierian Base in honor of Napier, the inventor of loga- 
rithms, and its value to thirty decimal places is 



2.71828 18284 59045 23536 02874 71353 



22 



SUMMATION OF DIFFERENCES 



15. Theorem. (i+-)^ 



Proof. Put -=z, then 



(-;)■ 



(i+s)i. 



Lt 



+^T=c^o(^+^)'=^- 



Q.E.D. 



16. Theorem. 



<7' — I 



^Xoz.a. 



Put rt'" = 5- + I. Then if v = o, c = o. We have 



log,(i+^)^= -log,(i +--) 



<?'■ — I 



Cor. 



log,(i + ^-)'' 

= loga ^. 



Lt 

?7 -1 O 



rt' — I 



Q.E.D. 



17. Summary. We have demonstrated the following impor- 
tant theorems : 



Lt tan,r ^ Lt _£ 
X = o 1- .r 



tan X 



Lt sm,r Lt ^ 



x= o 



Lt 



X 



l+" = 



sni X 
Lt 



I ; 



J =.rXo(l+'^)^=^5 



Lt rt'- I 



= log,^; 



Lt -1' 



^' — I 



Incr ^ ' 



C= I + I +~ + — -\ = 2. 7 1 828+. 

!2 ,3 



LIMITS 23 

18. Application to Analytic Geometry. Problem. To find 

the slope of the tangent through a given point of a curve. 

Let 7 =f{'i') be the equation of the curve and P^ = ('^'i»Ji) the 

given point. Take another point,* P^^ = (-^'2,72)' ^^ ^^^ curve. 

-1/ -1/ 

Then ^ is the slope of the secant Hne through Pj, P^. 

By taking /^2 sufficiently near to P^, the secant PiP^ can be 
made to lie as near the tangent through P^ as you please, and 
the slope of the secant can be made to differ as little as you 
please from the slope of the tangent ; that is, if 7^2 = ^lA then 
y^ = Ji, x^ = a:j, and we have 

^^ J') ~ y^ 
x\, = x^ -^ i = slope of tangent. 

Let the symbol A7 = J2 —fv ^^^ ^-^ — -^'2 ~ -^1 5 1 
then X2 = .t'l + A,r, j/g = Ji + Aj/, 

Lt J^2-Ji_ Lt Aj/ Lt /Ui 4- A.r) -/(^i) 



and 



Aa- = o ^^^^ A.r = o A;ir 



Therefore the slope of the tangent through the point (;r,j) 
of the curve 7 =f{x) is 

Lt Aj^ Lt /(.r + /i) -fix) 
^-^'=oA,r /''^o /i 

Let thrfe student draw the figure and review the proof. 

EXERCISES 

1. Find the equation of the tangent to the curve y = x- through the 
point whose abscissa is 2. Thus, Xi = 2, v, = 4. 

* Barrow's method for drawing a tangent, 

t Here the symbol = means literally " approached indefinitely near to." 
+ It is to be noticed that these definitions of Aj and A;r imply that Aj is a function 
of \x vanishing with the latter, y being also a function of x. 



24 SUxMMATION OP^ DIFFERENCES 

A)' (2 + A.r)- - 2- 4 Sx + Ax- 
SOLUTION. -/- = ^ -^ = ^ = 4 + Ajc. 

.Ax Ajv Ax 

^•^^ o A^^ "" ^^^^^^ °^ tangent = 4. 

Ans. (>'-4) = 4(jt"-2). 

2. Same as Ex. i for curve y = .r^'. Ans. j' — 12 .r + 16 = o. 

3. Find the tangent to the curve j- = ; at the point (a^i, ji'i). 



Solution. A^. + ^')-/(^.) ^ =h±]^Jll ^ -f^^. 
Hence the slope of tangent is :„ and the equation is 

X{ 

(J-J'i) =-Acv-A"i). 

4. Find the slope of the tangent to ,v = ^ at the point (^, ji')- 

Ans. r.. 

5. Find the slope of the curve r = ^^• 



Solution. ^t A, ^ Lt (^ + ^)^ .i^ l, .i (. + ^)^ 
Lt - I 



/^ = i, , 



_3 



6. Find the tangent to the curve r = — ^ at the point {xy,}\) 



Ans. y — y\^= — ^Xy ^ (.r — x^. 
^ind /^^^Q /(:^L±^)zl/M for the following : 



/ . h\ . h 



8. Find the slope of r = sin x. 

,. . 2 cosf j: + - Isin 

Aji' sni (.T + /^) — snijf _ \ 2J 2 

Ax h ~ ■ h 

Suggestion. Use a theorem of. the summary at end of Chapter II. 

Lt ^y 
Ans. . . ^-^ = cos X. 



LIMITS 25 



Lt Ay I 
9. y = \og^X, A.r = o^ = ^- 

10. y = a% ^]ri^^^ = a^\og.a. 



19. Notation. In the preceding problems our object was tp 

find the slope of a tangent, which was found by performing the 

• J- ^ 1 u Lt Aj/ , fix + h)— fix)' 
operation mdicated by ^^. ^ ^ -^, or by ^^-^ -^ — ^-^^ 

But this may be done whether we consider y=f{x) to be a curve 
or not ; and the result is, in general, another function of x, 
derived by this process and called the ''first derived fiinctioji,'' 
"'first differential coefficient,'' or simply the "derivative'' or 
" derivate" of the function. Thus the first derived function 
of f{x) is 

Lt f{ x-rJC )-f{x) 

and the first derived function of y considered as a function of 
X is 

A.r^o /^^-' 

these being the expressions which fully indicate the operations 
to be performed upon f{x) and y. We will also indicate this 

by the symbol — written before any function whose derivative 
dx 

with respect to x is to be found. Thus 

<^ / X d]> Lt Ar 

;^.W. or ^^-,t means ^..^„^. 



* Aj is by definition above the increment of y due to the increment A.r of x, y 
being also a function of x. 

t The notation dy and dx for the infinitesimal increments of y and x was intro- 
duced by Leibnitz (^Leibniz). 



26 SUMMATION OF DIFFERENCES 

Another notation frequently employed is explained by either 
of the definitions 

/'(.v).^./(x), </,'(x).,^t^i(£±^h:l(£). 

The slope of the curve y = ^jr(^x) may be expressed by any 
one of 

^, s ^ Lt ^|r(x-^/i)-^|r(x) Lt A^ ,, 



EXERCISES 

1. £(^ + 4)^ = 3(^ + 4)-^ j-/ = ^- 

2. y = sin ax. -■^ = a cos ^;c. 

3. f{x) = ^", ;z being a positive integer. f{x) = ;zjc"~^ 

4. — cos ax — — a sin ^Jt: ; — tan x = sec- Jir. 
dx dx 



;c— I (jc— i)2 

6. J' = l0ge^S. \\,\z)= ■ 

z 

7. — vf = ? — ;;z^ = ? 

8. If j^ := log c^(^), prove ^ = ^M. 

dx <f> {x) 
Solution. 

^i, r ^(x-Mv-^w /^ " I 

h \^ h ^{x)\ 

<t>(x) 

... ^' ^ *^'W . 



LIMITS 27 

9. Solve the first fifteen or more examples at page 50, Todhunter's 
Differential Calculus, by Fermat's method, indicated by 

Lt Ax^-h^-AxY 



h = o 



h 



20. The Differential Calculus. We have seen how, by apply- 
ing the formula -^— j ^-^, and, after certain algebraic 

reductions, passing to the limit of the fraction, we have been 
able to find the first differential coefficient of f{x). This pro- 
cess is called differentiation ; and when a function has been thus 
operated upon, it is said to be *' differentiated!' Now as there 
are but six fundamental algebraic operations and only a few 
transcendental functions in common use, we ought to be able 
to devise simple rules and corresponding formulae for writing 
out the differential coefficient of any such given form. For 
example, in our last lesson we found the derivative oi x'\ n being 
a positive integer, to be ;u'"~\ It can be shown that this result 
holds for all values of ;/ ; consequently we have the rule for 
differentiating any power of a variable with respect to that 
variable : " Take the product of the exponent and the variable 

luith its Exponent diminished by unity '' Thus: — x' = 7,1-^, 
d _-, --i d l^ -I -i ^ n 

■ — x ' = — 2x , — x^ = i'^' > Gtc, generally. 
dx dx 

These rules and the formulae expressing them in mathe- 
matical symbols constitute the differential calculus, t Hence, 
by means of a few simple rules we are able to tell what the 

* Strictly speaking, Fermat's method is one for determining the maxima and 
minima of functions of a single variable; but as it is so closely related to the funda- 
mental formula, /C-*" + —/\-^) ^ j.|^^^ Lagrange and other eminent mathematicians 

/i 
woulf] have given him credit for the invention of the differential calculus, it seems 
quite proper to associate his name with the general formula of differentiation. 

t Invented by Isaac Newton (1642-1727) and Gottfried Wilhelm Leibnitz (1646- 
1716) independently. While Newton certainly had priority, yet the notation of 
Leibnitz was as certainly superior ; and thus it is that the honors are quite evenly 
divided. 



28 SUMMATION OF DIFFERENCES 

result of applying the formula j^lj- l^-^^^-l is, without 

carrying out the indicated operations, which become in many 
cases exceedingly involved and difficult of execution. Herein 
lies the advantage of the differential calculus. 

21. The student should learn the rules and formulae immedi- 
ately ; and how to apply them. Each formula should be demon- 
strated independently of any other rule of the calculus, this 
being facilitated, in the case of the trigonometric, logarithmic, 
and inverse functions, by the summary of Art. 17: illustrated 
in the exercises of this chapter. 



CHAPTER III 

AREAS AND LIMITS OF SUMS 

22. We proceed to examine a method for finding the area 
enclosed by a given curve, the axis of x, and any two ordi- 
nates ; and thence a method for finding the limit of a sum of 
;/ terms of a certain form, when ;/ = oo. 

Let y=f{x) be the equation of the curve, f{x) being an 
increasing function in the first quadrant. Suppose we require 
the area enclosed by the curve, the ordinates f{a), /(^)» ^.nd 
the X axis. Suppose b> a. Then the base of the area is 
{b — a) in length. Divide the base into n equal parts and erect 
ordinates at the points of division. There will be {ii + i ) of 
these ordinates, y^, y^_^, ••• jk„+i, and their common distance apart 

will be ~ , which call A.r=//. Further, we can calculate 
n 

the lengths of these ordinates from the equations y^ =fia), 
f2 =f^ + ^0. '" yn =/(^ + n- I h)^f{b - h), j/„+i =Ab). 

If frorn the pqints of intersection of these ordinates with 
the curve, parallels to x be drawn in the positive direction to 
the next ordinate in each case, we shall form a system of ;/ 
rectangles, 

yi'i, yji, ••■ yji, 

whose area, y^h -\-yjL + ••• -\- yji, 

:s less than the required area ; each rectangle lying on the 
positive side of its generating ordinate. If the parallels are 
^Irawn in the negative direction, the area, j/2'^^ -^y^/'- + *** +J'»+A 
will be greater than the required area, and each of the 71 
rectangles will lie on the negative side of its generating ordinate. 
If f{x) be a decreasing function, the inequalities will be 
inverted. If f{x) be part of the time increasing and part of 

29 



30 SUMiMATION OF DIFFERENCES 

the time decreasing, or vice vcrsa^ the inequahties may or may 
not be inverted. In general we shall have inequalities ; thus 

X y^^^ > ^ > ^ yJ^ ' 

.r=l 2 

A being the area sought. We will call that system of rectangles 
lying to the right of their generating ordinates the system R ; 
the other will be L. Then 

^-^ yJ^y ^-% yJ^- 

1 2 

?H-1 1) 

Now L-R = ^ yji - ^ yji = (j|/„+i - y^h ; 

2 1 

that is, tJie difference between the area of the L system and the 
area of the R system is the same as the difference between the 
areas of the last L rectangle and first R rectaiigle ; a very impor- 
tant result. 

Let the student draw the figure and interpret the result 
graphically. 

23. By taking h small enough, and consequently n large 
enough, {y,^^i— y-^h = L — R can be made as small as you 
please. Therefore A, which lies between L and R, must differ 
from either by less than \{y„^x — y^li\. Therefore 

1 2 

Whenever one of these limits can be found, the result is the 
exact area required. 

Scholium. Since h is a common factor of the terms of 2J» 
we may write, 



2' 



24. The inequalities and reasoning above may be expressed 
in another way, employing the equations y-^^ =f(a), j'2 =/(^ + ^^\ 
...y^=,f{p — Ji), yn+\=f{^)j h = ^x, and a slightly modified 
notation, thus : 



AREAS AND LIMITS OF SUMS 31 

f{ayi-h/{a-\-/iyi + --+/{l^-2/iyi+/{d-/iyi=^f(,r)Ax=R,{a) 

f{a^hyi^-f{a^2hyL^''-^Ab-hyi-\-f{byi=^f{x)Lx=L. (b) 

This notation means that x is increased by ^x from term to 
term, from a to {b — A.r) in the first summation, and from 
{a H- Aa') to b in the second. This implies that {b — a) is an 

exact multiple of A.r; but that follows since = n, a posi- 

A,r 

tive integer by definition. It is not necessary, however, that 
this be so, nor indeed that the Aa-'s be all equal, for inequalities 
of the kind considered can always be devised by having 

i;A,r,<(^-^)<^A.r,. 

The question of incommensurability will not be more closely 
touched upon here. 

Hence from {a) and {b), 

But X~X- 1-^*^^^ -/(«)]^-'' vanishes with Ax. 
Therefore A = J-l ^Xf^x)A.x = ^,^1 „ t/W^"^'' 
by reasoning the same as before.* 

25. Let us apply the general reasoning to a concrete exam- 
ple. Take the curve y = -^^ ,r^ say, and let it be required to 
find the area between the ordinates where x = t, and x = 6. 
Use coordinate paper, drawing in the curve very nicely with an 
irregular draughting curve. The points are located most expe- 
ditiously by means of a table of squares. Adhering to our nota- 
tion, suppose we divide the base, (b — a), into three parts; then 
J'i = •9, J'2= ^-^^ ys= ^'S, fi= 3-6, Ax=/i^ i^ a=s, b = 6, fi=3. 
V x/i =s.o<A<y y,h = y.y. 

1 2 

* This method of quadratures, as explained in the last three articles, might be 
termed the modern method ofe exhaitstions. 



SUMMATION OF DIFFERENCES 



Hence the area, A, differs from y.y or from 5.0 by less than. 
(j/4 — J^i)/i = 2.^ units ; also seen from y.y — 5.0 = 2.7. 

Next divide the base into six equal parts ; then 71 — 6, 
A,r = h •■= J, and 






.900 
1.225 
1.600 
2.025 
2.500 

3-025 
11.275 



.•.;^= 5.6375 and ^ = 2^ - V 4- JV/^ = 6.9875. 

Hence 5.6375 < ^ < 6.9875, and the error is less than 1.35 
units, a closer approximation than before. 

Again, take I\x=h = .i. Then from a table of squares we 
readily find the numerical values of the consecutive ordinates 
beginning with the second to be as follows : 

21 31 



Xy-^ 





961 




024 




089 




156 




225 




296 




369 




444 




521 




600 


12 


685 



1.68 1 


2.601 


1.764 


2.704 


1.849 


2.809 


1.936 


2.916 


2.025 


3.025 


2. 116 


3-136 


2.209 


3-249 


2.304 


3-364 


2.401 


3.481 


2.500 


3.600 



20.785 



30.885 



AREAS AND LIMITS OF SUMS 



33 



and 



■)i 11 ji oi 

2 2 12 22 

30 ;ji 

2^ = I^-jV^+V^ = 6.i655, 



Hence, 6.4355 >^ > 6.1655, ^^^ the error made in taking 
either number for A is less than 0.27 units; shown also by 

(j3i-/iy^ = 0-27. 

Of course it is easy to see now, from (j/«+i —y-^)h, that for 300 
subdivisions of {b — a) the error would be less than 0.027; foi" 
3000 subdivisions less than 0.0027 ; and so on. Let us see if the 
limit can be found. 

26. Introducing the literal values given in the example, we 
have ^ ^ 

x=--« a-i-Ax 

that is, 

A.r 



A< 



A< 



10 

Ax 
10 

Ax 
10 

Ax 
10 
b — a 

10 71 

ion 
Whence 

b — a 



a^-^{a + hf-^'" + {a + ji-ihf 



<A, 



n-l »-l 



iia- 



^2ah^z^JP'^z 
1 1 

na^ -\-2a 



na^ 



<A, 



1 1 

b—an{ii~ i) (b — a\-{n— i)n{2 n— i)" 



<^, 



b — a 7i{ii-\- 1) fb — a^ 7i{n+ i)(2;/+ i) 



10 



b-a 



na^ -\-2a 



a^ + a(b-a)[i-~) + {b-a) 



10 



a'^ + a(b-a)fi-\-^) + (b-af 



(-,;)(-,;)] 



<A, 



>A. 



34 SUMMATION OF DIFP^ERENCES 

The first members of these last two inequalities are the areas 
of the R and L systems of rectangles in terms of the number 
of rectangles. Whence, making n equal to 3, 6, 30, and <? = 3, 
^ = 6, we derive in order, 

7.7 >^> 5.0, 
6.9875 >y^> 5.6375, 
6.4355 >^>6.i655; 

which agree with the results just calculated in another way. 
Taking n larger, R becomes larger and L smaller ; by taking n 
large enough, both R and L may be made to differ as little as 
you please from 

T-V \_a\b -a) + a{b-af + l{b - af] ^^-~- 

f b"^ a^\ 
Therefore the area required cannot differ from =6.3, 

for the data of the problem. It should be noticed that, since no 

fb^ a^\ 
restriction was placed upon a and b, the formula ( J is a 

general expression for the area enclosed by the curve -^-^ x^, 
the axis of x and any two ordinates whose abscissae are a and b. 
The result of the investigation is indicated by 

A;t = o ^ 10 Ax=o A^io 30 30 

27. The Symbol f.* We indicated A,^ln^ by ^; then 
let us indicate 



h±\x 

A;r: 



a±\x 

Hence we have the equation 
•6 ,,2 



Ju 10 30 3c 



A 

30 



This symbol, the long s, was introduced by Leibnitz, and stands for "lij?iit of sum." 



AREAS AND LIMITS OF SUMS 35 

28. It was shown in the preceding part of this chapter how 
the solution of the problem of areas can be made to depend 
upon the solution of the problem of a limit of a sum. Many of 
the most important problems in mechanics can be made to 
depend upon a limit of a sum. We will take a simple example. 
Suppose we wish to find the space described in a given time by 
a freely falling body. We know by experiment that the velocity 
gains a definite number of units in each unit of time. This 
gain per second in velocity is called the acceleration of the body, 
and is represented by^. Near the surface of the earth ^'-=32.2, 
in the foot-pound-second system of units ; this meaning that the 
velocity increases 32.2 feet per second in a second. 

Solution. Let 7^ be the velocity at the beginning of the given time- 
interval, and measure time from that instant, representing it by 0. 

Divide the total interval, /, into ;; equal parts, each - = \Q, and 

n 

represent the elapsed time at the beginning of the rth partial interval by 
0, = r — 1 A^. Then the velocities at the beginning and end of the rth 
interval are V(^-\-gO^, Vo+gO,.+i. 

Whence ^(7>q+ gO,)A6 < s <^(7\^ + gO,.)^0 ; where s is the re- 

7-=l r=2 

quired actual space passed over during the time /. 



But ^ (r'o +gOr)^e - ^ {7'o+g0r)A0 



x-x 



OAO 



=g{o^^+i-o,)/^o=gt\e 

vanishes when A^ = o, from which it follows that 

r=l »•=! ,-1 



36 SUMMATION OF DIFFERExNCES 

and, resorting to the method of summation in Chapter I., using the 
identity 

(x + Ax)- X- Ax- 
XAX = , 

2 2 2 



Lt 



ot«.^^ = A«1o^«^^ 



Lt jl 

A^ = O j 2 



r t t-\') n 



- A6» 



X-X ^'-IX^^' 



A^ = 0|2 2 2 J 



Lt f /- z:^^ 1 _ /■' 

A9 = o 



^ ^ 

1^ 2 2 J 2 



2 



the well-known formula for the space passed over by a freely falHng 
body with initial velocity z'q. 

29. Attraction. We know that the sun, the planets, and 
other celestial bodies attract one another ; and the attraction of 
two leaden balls has been approximately measured. 

Newton's Law of Universal Gravitation may be stated as 
follows : A71J/ two material particles in the universe attract each 
other luith a force wJiicJi vaides as the product of tJieir masses 
directly and the square of their distance apart inversely. 

For analytical purposes we may explain that material particle 
means indefinitely small body, and that distance apart refers 
to a distance indefinitely great as compared to the linear dimen- 
sions of the particles. So that all parts of each particle are 
supposed to be at the same distance from the other particle ; 
or, to put it still more mathematically, we are to suppose it 
possible to accumulate attracting matter at a point. 



AREAS AND LIMITS OF SUMS 37 

30. Problem. Required the attraction exerted by a very 
thin homogeneous straight rod, whose mass per unit length 
is S, upon a very small mass ;«' in the produced axis of the rod, 
distant a units from the near end and b from the remote end. 

Solution. Let the line MN represent the rod, and ;;/' the small mass 
in its axis, taken so that J/;;/' = a and Nm^ = b. Consider a portion 
of the rod i\x in length, and let x be the distance of m\ from the nearer 

end of Ajc". Then 8A;c is the mass of the portion, and k ; — would be 

x' 

the attraction between A.r and ;;/ if all the mass, 8Ajc, were concen- 
trated at the distance x from ;;/. Thus, if a^ be the attraction of the 
masses considered, we shall have 

;>;/'8Aj:^ ^ , w'SAx , ^ . 

k -~ > ^^ > /& ' k—d. constant. 



X' ix -\- Ax)- 



\ i^X ^ A ^ 7 CV f X. A I^X 



where A is the required attraction. From this, by taking the difference 
between the extreme members of the double inequality, we easily show 



that A = km 



6 1 -^- It remams to rind I — ;;• 

*,/« X' J a Jt" 



From the identity ^ = ^ \ — ^ — , we have 

^) ■' x:^ X X -\- h x-{x ■\- li) ■ 



-?/— As '' n ''— '« 



/r^ 



a L. a o+A-c— I a \ ' / 

h-n 

a b ^.r-(.T + /0 



I I , ;o 

a b 



where p is a finite quantity, the arithmetical mean of the n terms of 

6-ft 

^ —■ ; a quantity which lies between and -, 

^x\x + h)' ^ ^ a-{a^h) {b-hyb 



limits — and — . Also nh = 7?/lx = (b — a). 



and consequently, for any admissible value of Ajc = /i, between the 
— and — 

A = km'Sr'^^=h;i,8.\'\^-'j+p{b-a)/i\=/cm's(^-])- 
J a X' "■ ~ ^ { a b ) \a bj 



38 SUMMATION OF DIFFERENCES 

31. From the foregoing examples we gain some idea of the 
utility and importance of the notion of a limit of a sum. The 
following problems will illustrate further : 

EXERCISES 

1. Let the student solve the example of the area of the curve ^^ x- 
by finding the sums in terms of Ax instead of ;/, using an identity 
which he should be able to form for himself. 

2. Solve the example of the falling body by the method of the exam- 
ple of area. In this form it involves an arithmetical progression. 

Z^x = o ^ 4 4 4 

wliere Ax = k, letting p be a quantity which lies between the greatest 

and least of the n terms of V ^' 't~ 4^ , 

^ 4 



X 

J a X^' 



Ajis. I x^dx 

4 4 

•6 

4. Find 



Solution, j^ ^ = ^.^oS ^ = A.^o S ^^ = A. ^o X ^' 

a + Aa; a ar^Syx 

Xb ^j^ ^ \ „ 

a 

Ax I I ('ix-\-2^x)Ax' 

Also — x = . -, , ■ v' + ■;/ , . v> • 

X' 2 X- 2 {x + Ax)- 2 X {x -\- Ax)- 

. r^_ Lt ^^_ Lt |_t I I . ■>, 1 

where p and Ax;/p = (/ + Ax — rt:)p are finite quantities, ;/ being the 



number of terms in > ^^^ 7" —^' Thus Ax-/ip vanishes when 

^ 2xVx + Ax)2 '^ 

Ax = o. ^ ^ 

Therefore I -— = — ^ -. 

. Ar x" 2 ^- 2 ^- 



AREAS AND LIMITS OF SUMS 39 



►6 
COS X{/X = ? 



Use cos jcAx = \ sin I x + - — sin ;c ]\ 



1 \ 2/ V 2j \ . h 

^ sin - 

2 

y^;/j-. sin ^ — sin a. 
Lt ^Ax 



Lt V - ? 



/ 1- 
Solution. W e know from algebra that log ( i + j') = j' \- R, 

where, if ;- < I, i? < — < — • 

3 2 

Hence, 

log i+— +- -^ >-_>iog— I + -( — -- — 

V X J 2\X J X ^ X 2\X J 3 V -"^V 



and 



^ , , Aa- ^ Aa^ '^ Ax ^ , A.v v:^ Ix Aa" ^,r:^ Ajc 

a a a a *-' 

But > I02 = loCT , Lt > — ,,- = -,, 



Lt > — - = ,, j:,, Lt — = and Lt = o. 

^^ X' 2 a- 2 0- 2 3 

Therefore | -'^^^rloe-. 

7. Let the student see if he can name the most important difficulty 
which arises in the solution of these problems. 

8. Explain the meaning of the equation 

and tell why it is that any finite number of vanishing terms may be 
omitted from the summation without affecting the limit. Illustrate 
graphically. 



CHAPTER IV 
OBSERVATIONS UPON THE INTEGRAL CALCULUS 

32. Before taking up this chapter the student should have 
acquainted himself with the rules of differentiation and the forms 
of ordinary derivatives. He may have observed in the prob- 
lems of summation, that to find the limits of sums of the form 
V(^(,r)A;r, it was necessary, in the general case, to have an 
identity of the form 

(l){x)Ax-~f(,r)- ylr{x + Ax) -f F{x, A,r)A,r2. 

The fundamental theorem of the integral calculus puts into 
mathematical language a rule for finding the limit of any sum, 
of the kind considered, provided an identity of the right form 
can be found ; and the rules and formulae of the integral calcu- 
lus afford a method for the discovery of the essential form of 
the identity when it exists. 

33. Fundamental Theorem. rf(,v)dv=/{d)-/(a). Or, 
more explicitly, 

$■ where '^'{^') is any function of a' and ^/^(.i') any function whose 
differential coefficient with respect to ,r is '^'{-r). 



<k'i^) 



Proof. We know that 

Lt c{>(x-j-/i)-<t>{x) 

/i = o k 

hence 

<^{x^h)-<^{x) , _ ,,. . 
J Art=^ {x), 

where e is a quantity vanishing with // ^ Ax. 

40 



OBSERVATIONS UPON THE INTEGRAL CALCULUS 41 

Thus cf>'{x)Ax=cf>{x + /i)-(l>{x) + /i€ 

= <^(/.) -<^(^) + /ij J^ + /^ - a)« 

where n is the number of terms in the summation and a the arithmeti- 
cal mean of the n terms of the type e, a quantity vanishing with h. 
Therefore the theorem 



jy\x)dx=f{/^)-f{a) 



Q.E.D. 



34. The importance and application of the theorem will be 
shown immediately by using it in the solution of the problems 
already solved. 

In finding the area of the parabola iV-^'^' ^^ ^^^ shown that 

the area = | '^dx. But — is a function whose derivative is 
^2 4. 10 30 

Therefore, by the theorem, 



10 



b'' d 



Jr^ x^ 
I ~dx 
« 10 30 30 

In the case of the falling body the space passed over was 
shown to be I {z\^+^^0)d9. But ^)^+^— is a function whose 

c/o 2 

derivative is Vo+g-9. Therefore 



/2 



2 
In the next problem the attraction was hn'Sj ~ ; and 

7 't X 

is a function whose derivative is — „ • Therefore 

x"^ 

^ = .,«'3j(-i)_(-l)j=..'3(i-i 



42 SUMiMATION OF DIFFERENCES 

The remaining problems are tabulated in order in the three 
columns below, the first giving the problem, the second the 
function whose derivative is to be of a given form, and the third 
the result : 



a 

i 
J, 



4 4 T 
''dv I I I 

X^ 2 X^ 2 d^ 2 b^ 

b 

cosxdx, sin;i', sin Z'- — sin<^. 



■"dx 

X 



log X, log d - log (7. 



35. The expression i <p(x)dx is called the definite integral 
of (\>{x)^ or o/(f)(x)dx, betivecn the limits a and b. No\v the funda- 
mental theorem tells us that the value of this definite integral can 
be found by finding the function whose differential coefficient 
is ^{x\ substituting first b, and then a, for.i'in this function, and 
subtracting the latter result from the former. We then very 
properly define 1 ^{x)dx, to mean any function of x, which, 
when differentiated, gives <^{x'). Thus, let -^{x) be such a 



function ; then — | -^{x) + c 



~-ylr(x) = (f)(x). Hence there 



are an infinite number of such functions, any two of which 
differ by a constant, but any one of which satisfies the 
conditions of the theorem. We call 1 (l)(,r)dx the indefinite 
integral of (t^{x); it is a function which, when found, generally 
leads to the definite integral between the given limits. The 

I'O (X 

indefinite integrals in the above problems are '—^e, v^^ ^'^Gr--\-c, 

I f* I 

V c' — V c, s + <f, sin X + c, and los: x -^ c. Hence we 

,r 4 2,1-2 to 

I F{x)dx is not a function of x. What is it a func- 

tion of .-^ -'^^ I ^(-^')^'^' ^ function of ,r ? 



OBSERVATIONS UPON THE INTEGRAL CALCULUS 43 

36. The indefinite integral does not always exist for every 
form of function taken for derivative, and no general rule for 
finding it can be given without the use of transcendental func- 
tions. Thus the processes of the integral calculus are indirect, 
and depend upon our knowledge of the forms of derivatives 
gained in the differential calculus. We mean that the integral 
calculus depends upon the differential calculus, but that definite 
integrals, and consequently indefinite integrals, may be found 
by the ordinary processes of algebra without reference to differ- 
entiation ; if this be called integration, then integration does 
not necessarily depend upon differentiation, a point very impor 
tant to understand. 

37. After reading the integral calculus, the student should 
follow it up with references to the Encyclopcedia Britannica, 
at "Infinitesimal Calculus" (Historical Introduction) and yt 
"Archimedes"; Cajori's History of MatJiematics, at " Hippias 
of Elis," " Antiphon," " Eudoxus," and elsewhere. Thus it 
was that the cr^rm of the infinitesimal analysis had already 
assumed tangible form even in the remotest of antiquity. 



SEVEN LESSONS 



IN THEORY OF 



INVERSIONS OF ORDER 



AND 



DETERMINANTS 



BY 



B. F. GROAT 

Assistant Professor of Mathematics and Mechanics in the 
University of Minnesota 



MINNEAPOLIS 

H. W. WILSON, Publislier 

1902 



THE LIBRARY OF 

CONGRCSS, 
Ty»o Copiee Reobvco 

FEB. t? 1903 



|DLAS«a_XXa Ho. 



corr B. 



COPYRIGHT, 1902. 

BY 

B. F. GROAT 



PREFACE 

No student should enter upon the study of analytic geom- 
etry without an elementary knowledge of that part of the 
theory of determinants which treats of elimination and the so- 
lution of simultaneous equatioiis. This will be the more readily 
conceded when it is shown that such knowledge may be gained 
in fewer than a dozen lessons. Seven introductory lessons in 
theory of inversions and determinants would scarcely suffice 
per sc to make a lasting impression upon the student; but 
if the elementary principles thus learned are immediately ap- 
plied to the solution of problems in analytic geometry the 
good gained from such a brief course should be very consider- 
able. In whatever manner the student be introduced to a sub- 
ject, it is, in most cases, only by constant recurrence and many 
varied illustrations that the value of a process is finalh^ realized. 

It is thought by the writer that the treatment of inversions 
as applied to determinants is new in some respects, and atten- 
tion is invited to the /z2/;/^^r^<^ theorems and related matter 
which form a chain of reasoning leading up to Laplace's de- 
velopment. 



University of Minnesota 
April, 1902. 



INVERSIONS AND DETERMINANTS 

CHAPTER I 
INVERSIONS OF ORDER 



1. If several consecutive symbols of* a sequence, as letters 
of the alphabet, or numerals, be written in a line in any order, 
then an inversion of natural order, or simply an inversion, oc- 
curs whenever any symbol follows another which it should 
//^///r^?//;/ precede. In bac, 5647, 4321, JKIHG, 564213, there 
are respectively i, 2, 6, 9, 12 inversions. 

2. In considering any such line of symbols we shall number 
their positions to the right beginning at the left. Then the 
order of position of an}^ symbol in the line is of the ist, 2nd, 3d, 
degree according as it occupies the ist, 2nd, 3d, position; and 
so on. ■ The order of a symbol, and the degree of the order, 
is taken with reference to the natural sequence of the sym- 
bols in the line: it is the same as the order of the position 
it would occupy if the symbols were arranged in natural order 
in the positions of the line. It will be frequently convenient 
to refer to a symbol, or position, as being even or odd, meaning 
that the order of the symbol, or position, is of even or odd 
<!cgree. 

3. Theorem I. In any line of symbols capable oj arrangeincnt 
in natural sequence, if any tivo symbols be vitercha?iged the number 
of inversions in the line will be increased or decreased by an odd 
jiumber. 

The interchange of any two symbols occupying 



6 INVERSIONS AND DETERMINANTS 

positions obviously causes a loss or gain of a single inversion. 
If there are m symbols intervening between the two to be 
interchanged, then the transfer of each of the two to the posi- 
tion of the other obviously causes a loss or gain of one in- 
version with each intermediate; moreo\'er there is also a loss 
or gain of one inversion between the two symbols interchanged. 
Therefore there are 27n^i^ an odd number, of losses and gains 
together. But the change in the number of inversions is the 
differeJicch^\.^N^^x\ losses and gains, and if the sum of two in- 
tegers be odd their difference is odd. Therefore the change i;i 
•he number of inversions is odd in any case. 

4. A complex symbol can be formed by uniting into a single 
symbol of two simple parts any two symbols chosen from as 
many different sets of sequences of the kind we have been con- 
sidering. A triply complex symbol maybe formed by choosing 
from three sets of sequences; and so on. As illustrations, 

(11), • , h,A^, Q\ ci- B' are doublv, and A- m^^.'^X , triplv com- 
' X ' y' ^ ^ ' 

plex. We shall confine our remarks to the first kind. 

5. Definition. The order of a complex symbol is the sum 
of the orders of its simpleparts. 

6. Theorem II. If to any number of cojiseciitive letters taken in 
any order y the same number of consecutive numerical suffixes be at- 
tached, one suffix to eacJi letter y then upon writing these complex- sym- 
bols in line in any order at pleasure, the total nuinber of inversions 
among both letters and suffixes ivill be either ahvays odd or ahuays 
even. ^ 

In any one interchange of two symbols the number of in\'cr- 
sions among either letters or suffixes is changed by an odd 
number (Theorem I); and the sum or difference of two odd 
numbers is an even number. Hence the total number of in- 
versions after one interchange remains either odd or even as 
it was. But by successive interchanges two at a time the sym- 
bols can be brought into any prescribed order one at a tim? 
The truth of the theorem is apparent. 



INVERSIONS OF ORDER 7 

Example. The number of inversions in a\b-2c':\(fj^, (fAL:.a\b2, 
d\Czb'2a\, or any other permutation of the group, is resjiectively 
o, 10, 12, even. 

Scholium. The greatest possible number of inversions in 
such groups is n{ii — i), where n is the number of complex 
symbols. 

7. Theorem III. In any Hue of doubly complex symbols, i^Jiosc 
letters and suffixes arc the incmbcrs of correspo}idi)ig sequences, the 
total nn/jiber of inversions due to the presence of any specified com- 
plex symbol, H^y is even or odd according as the order of that sym- 
bol is ei'cn or odd. 

If the order of Hx is even, then H and x are both e\en or 
both odd; consequently there must be present an even number 
of letters and sufifixes together which are of lower orders than 
77, X respectively. If H^ is odd, then //, x are one odd the 
other even, and there must be present an odd number of 
letters and suffixes together of lower orders than //, x re- 
spectively. Therefore the number of inversions due to H^ in 
the ^/'.T/ position of the line is even or odd according as Hx is 
e\en or odd, since the only inversions due to Hx in that posi- 
tion are with letters and suffixes of lower orders than H, x re- 
spectively. But the number of inversions due to Hx is always 
e\en or alwa^'s odd, independent of the arrangement of the 
line,' since in any one interchange, and consequently in any 
succession of interchanges, it is impossible for Hx to gain or 
lose an odd number of inversions with any other doubh^ com- 
plex symbol. The theorem follows. 

Cor. The number of inversions due to the symbol Hx is even or 
odd according as ( — ly is -j- or — ; r being the order of Hx- 

Cor. If the7'e be a line of simple symbols arranged i>i any 
order, then the nitmber of iiiversions in the line due to the pres- 
ence of any specified symbol of the sequence luill be even or odd 
according as the sum of the orders of the specified symbol a7id 
its position is even or odd. ■ 

For a line of doubly complex symbols may be written with 
its letters in natural order and its order of suffixes correspond- 
ing to the order of symbols in the given line. Then the order 
of position of any suffix is the same as the order of its literal 
partner, and the number of inversions in the given line is the 



8 INVERSIONS AND DETERMINANTS 

same as the number in the written line. The corollary follows 
from this and Theorem III. 

8. Definition. The order of a group, of two or more com- 
plex symbols, is the sum of the orders of the constituents of 
the group. 

9. Theorem IV. The number of inversions in any line of 
doubly complex symbols y due to the presence of any specified two or 
more such symbols, is even or odd according as ( — i) ^ + Hs -^ or — ; 
s bci)ig the order of the specified group andi its number of inversions. 

Let ^ be the sum of the numbers of inversions respectively 
due to each symbol of the specified group considered sepa- 
rately. Then ^ includes each inversion occuring among the 
specified symbols twice, and each inversion between a symbol 
within the group and another without the group b/.t once. 
Therefore S — ns the number of inversions due to the specified 
group in the line. But 5 and s are both even or both odd 
(Theorem III and properties of numbers). Therefore the 
number of inversions due to the group is e\en or odd accord- 
ing as S — /, and consequently as s — i, is even or odd; that is 
according as { — 1)''~', and therefore as ( — i) *^% is -^ or — . 

Cor. Theorem III is a special case of The ore in IJ\ since then 
s ^=r and i = 0. 

Examples. 

I, Is the number of inversions in a^. c\fi gi& d%e^bh, due to 
the presence of a-i ^e dz hr,, even or odd? For ease of enumera- 
tion represent the groups by ^ 1 \ S — ^.^^ — 1 4 

2176345 2635 

In the latter {s — z)=30 — 5 = 25, and ( — \)-^ is - . Hence there 

should be an odd wwrvih&x of inversions due to ^ -1^ — ; and by 

2635 

actual count tliere are 17 in — 5 7_4_i and 2 in ^ ?, leav- 

2176345 174 

I 7 4. 2 
ing 15, an odd number, due to ' "^ 



2635 
I 7 4 



In Ex. I s — 2=24 for-^i ?-; and accordingly there are 



12 inversions due to ^ ^, 

I 7 4 
3. In d\ a-2 c\ bh en there are 6 inversions due tt) the presence 
of 64 d':! (/i for wlilch .9—/= 16 an <?7'i7/ number. 



CHAPTER II 
ARRAYS AND DETERMINANTS 



10, The expression 



12 


4 


—71 


6 


51 


32 


-16 


31 


17 


22 


4 


2 


9 


I 


— 8 


17 



exhibits 4'^=i6 



quantities included between the verticals. In general an ex- 
hibit of 1? quantities may be formed in square array by ar- 
ranging them thus in n vertical columns and n horizontal rows. 
Three exhibits in square array of the 9 quantities i, 2, 3,. . .9, 
are 

147; 123 159 

258 456 267 

.369, 7891. 348 

11. In any square array Vv^e will number the columns in con- 
secutive order to the right beginning at the left, and the rows 
in consecutive order downward beginning at the top. Thus, 
the third column, or column 3, in the first array above contains 
the quantities — 71, — 16, 4, - — 8, while row 2 contains 51, 32, 
— 16, 31. Instead of numbering the columns and rows i, 2, 3,. . ., 
it will frequently be convenient to letter them a, b, c,. . . ; i.e. 
column 3 = column c, etc. 

12. With two sequences of 71 simple symbols each ;2^ differ- 
ent complex symbols can be formed by writing every symbol 
of one sec]uence ?^ times and affixing in any uniform manner, 
to each one so wTitten, the 71 symbols of the other seqence, 
sint^ly, in succession. In illustration, the 3-=9 symbols a\,ai,a>„ 
b\, b-2, b-6, c\, c-i, a, are composed of the three letters a^ b, c, and 
the three figures i, 2, 3. Again, the nine complex symbols 

I 2 3 I 2 3 I 2 3 

't "t 't 9 7 9 ^j i i o^ th^ ^^^^^^ (i^)' (^2), (13), (21), {22), 

(23)' (31). (32), (33)» are formed from i, 2, 3, and i, 2, 3. 



INVERSIONS AND DETERMINANTS 



Hence we may represent any li^ quantities in square array by 
any one of 

av a^i a%....an a\ h c\ m\ (ii) (12) (13) {\n) * 

b\ b-2 h . . . . hi a-i bi c-j. . . . tm 



C\ Cl C-B 



.c,, 



a-i bz (f 3 . . . . in% 



,/;/! in-2 77U . . . m^, j, \ an bn Cn . . .Ifln 



(21) (22) (23) (2;/) 

{31) (32) (33).... (3^0 

{ill) {712) (;^3).. ,\7m) 



I 


2 

I 


3 


n 


T 


I 


1 1 


I 


2 
2 


3 


« i 


2 


2 


2 1 


I 


2 


3 


« 


^ 


3 


3 


3 


I 


2 


3 


n 


K 


;z 


n 


n 



13. In the first array just written the order of any letter is 
the same as the order of the row^ containing it, and the order 
of any sufBx is the same as the order of the column containing 
it. In the second array letters correspond to columns and suf- 
fixes to rows. The modes of arrangement in the remaining 
two arrays arc apparent. It is e\idently immaterial w^hich one 
of these, or other similar forms, is used, simply for the pur- 
pose of rcprese7iti7ig a square array. 

14. Definitions. The ;z- quantities of an array are called 
constituents, or elements, of the array. The order of a constit- 
uent with respect to rows or columns is respective!}^ the same 
as the order of the row or column containing it. The order of 
a constituent with respect to rows a7id columns is the sum of 
the orders of the row and column containing it. A square ar- 
ray of 71 rows and 71 columns, and consequently containing ir 
constituents, is said to be of the ;/th order. Care must be 
taken to distinguish different classes of order. 

15. Hence we may not only represent an array of quanti 
ties by a similar array of complex symbols, but each symbci 

* Here ot course the fi^^ures. and the numbers resulting from their combination, do 
not have" their usual sijjnification; but any symbol, as - or (32"). represents the 9z<a«/r(>' which 
stands in tiie same position in the array represented. 



ARRAYS AND DETERMINANTS ii 

may be taken to represent the value div\d position of the corres- 
ponding quantity of the array represented, by having the orders 
of every pair of associated simple components correspond to 
the two orders of position referred respectively to rows and 
columns as has been explained. 

i6. Coordination of products. The product of any n con- 
stituents of a sqv.are array of the ?/th order, so selected that 
one and only one constituent is taken from each row and one 
and only one from each column, we shall refer to diS di proditcf 
of the array. There are other ways of . coordinating products, 
but this is the only one with which we shall be concerned. 

17. The diagonal row of constituents through the top left 
hand corner of the array is called the principal diagonal, and 
the product of the constituents of this diagonal is the principal 
product of the array, 

18. In considering any such product, take the factors in 
order to the right, beginning at the left. Then, corresponding 
to this order of factors ^ there is an order in which they were 
taken from columns and an order in which from rows. We 
shall speak of such an order as the order of cJioice^ or oi'der of 
selectio?!, from columns or rows as the case may be; when such 
choice has been made in the natural order of rows or columns 
we shall refer to it as the Jiatttral order of choice. 

Illustration. What is the order of choice from rows and col- 



umns in the product,^ b I e,oi llie arra; 



a b c d 

e f ^ h 

i j ^k I 

;ji n p q 



The order of selection from columns is 3241, or CBDA;Xh.c 
order of choice from rows is 4132, or DACB\ the order of 
choice from columns a7id rows is represented by c^bidsai, or by 
3241 . 
4132 

19. Theorem, hi any prodiict of an array , coordinated as in 
Art, j6, the total number ofinversio?is of natural order in the orders 



12 INVERSIONS AND DETERMINANTS 

of clioicc froni7'oics and coliunns is nhvays even or ahuays oeld ir- 
respective of the order of factors. 

The i^eneral square arra}' ma}- be represented by 

III I ' 

123 H 

2 2 2 

^ ^ 3 

-5 3 3 

I 2 3 



the natural orders of superior and inferior components corres- 
ponding respective!}^ to natural orders of rows and columns. 

Take any product of the arra}% as a ,8 7 o-, represented b}' 

[y, • • ^; ; then the orders of arrangement of superior and in- 
inferior components, corresponding to the order of factors, is 
the same as the orders of choice from rows and columns re- 
spectivel3\ But (Theorem II) the total number of inversions 
among the complex s3mibols in the product is always even or 
always odd irrespective of the order of factors. Therefore, the 
total number of inversions of natural order, in the order of 
choice from rows and columns, is always e^■en or always odd 
irrespective of the order of factors. 

Cor. If tlie sign- factor [^ — if be attached to any product of 
the kind mentioned, i being the total 7ni))iber of inversions of 
natural order in the orders of clioice from roius and columns, tJioi 
the sign iiith which the product is to be ultimately affected is alu^ays 
— , or ahvays' — , irrespective of the order of factors. 

_;o. The determinant of a square array of the nt\i order is 
the algebraic sum of all the possible products of the ir con- 
stituents taken n together, limited by the condition that one 
and only one constituent is taken from each row and one and 
only one from each column; the sign with which any product 
is ultimately affected being -V, or — , according as there is an 
even or odd total number of inversions of natural order in the 
orders of choice from rows and columns. 



21. The determinant of a square array is frequently repre- 
sented by L\. Hence the definition of a determinant, ex- 
pressed in mathematical symbols, is _\ = 2'( — i)'{aib2CB m„); 



ARRAYS AND DETERMINANTS 



13 



where )i is the degree of the array and i the total number of 
inversions in the term to which it refers; the summation to ex- 
tend over all the possible terms that can be brought into co- 
ordination by the rule of Art. 16. 



ai bi 



. nh2 



will be represented by 



22. The square array 

I a,, bn m,^ 

\ai b2. . . .mn\c, it being understood that /^i, A2, .///«, are but 

the constituents of the principal diagonal of the array. The 
suffix c denotes that the superior components of the complex 
symbols correspond to columns. In like manner 



ai a-i. 
b\ b-i . . 


..^« 
..^. 


mi ;;/2. 


../;/„ 



will be represented hy\aid2 mn\n ^ referring to rows in 

the same manner that ^does to columns. 



23. When the terms of a determinant have been written out 
with their proper signs, or when the process of writing them 
out according to any definite rule has been represented, the 
deterriiinant is said to be developed. 



CHAPTER III 
FORMATION OF A DETERMINANT 

24. In the sequel an array will be frequently spoken of as 
being a determinant, meaning however the determinant of the 
array; just as in algebra x frequently means value of x. In the 
same way it will be convenient to refer to rows and columns 
of a determinant, meaning rows and columns of an array from 
w^hich the determinant may be de\'eloped. When no distinc- 
tion is necessary the word line will be used indifferently for 
row or column. 

25. There are n! terms of A. Let n,^2, ?'„, represent 

the n rows and c\,Ci,. . . .c^ the 71 columns of a determinant, £^, 
of the ?/th order. By (r;f„<f;t,) = ^, we shall mean the constitu- 
ent at the junction of the hth. row and /^th column. Associat- 
ing any r, as r/., with any <r, as Ck, we shall have, ^', a constit::- 
ent of A- Associating any one of the remaining r'^, as n-, \\\l\\ 
any one of the remaining cs, as ^/„ another constituent, •, is 
represented which is not in the row or column of •'. Ar>\un, 
associating any r still remaining, r,-, with any c still remaining, 
Ct, we have J, a constituent not in any row or column already 
chosen. Continuing this process until all the rows and col- 
umns have been exhausted, we have a set of constituents co- 
ordinated by the rule of Art. 16. The product of these con- 
stituents together with the sign-factor ( — i)', i being the num- 
ber of inversions in ^ | ^ • — , is, by definition, a term of A^ 
it is clear that any permutation of n, r-j. . .r,i combined with any 
permutation of ci, C2. . . .c„, one r wath one Cy will lead to a term 
of „. But there are ?i\ permutations of 7t symbols taken all 
at a time. Hence there are (?^!)^ ways of writing terms of 
A, and no more. But each term may be written in n\ ways. 
Therefore there are (;z!)^ -f- /z! =«! terms of A, and no more. 

26. It is important to bear in mind that such expressions 
as^^j • • ^, where the superior components refer to rows and 



FORMATION OF A DETERMINANT 15 

the inferior to columns, or vice versa, fully indicate the orders 
of choice from rows and columns and represe?tt ih.^ constituents 
of a product. 

Examples. 

I. What are the signs of the terms z /; q (^^^ lee n,fk in d, 
of the determinant of i ^ ^ ^ (j 

e f R h \^ 
i j k I { 
m n p q \ 

The foil wing scheme shows the method of determining 
signs : 



term: 


i b q ^ 


/ c c n 


/ k m d 


order of \ rows: 


3142 


7/3 a 5 


2341 


selection i columns: 


1243 


da c b 


2314 


number inversions: 


4 


7 


5 


sign of the term: 


-r 


— 


— 



Scholium. The factors of the terms may be rearranged in 
natural order as to rows or columns. Then the process maybe 
as under: 

i h i]^ q c e I n d f k 7n 

3124 C A D B 5 fi y a 

23 5 

Of course the same signs result as before. 

2. Is e p j c a term of A in Ex. i? 

27. Let \a\ b-i....m„\. represent the determinant of a 
square array of the «th order (Art. 22) to be developed. Then 
selecting constituents from columns in natural order for every 
term, permuting into new order the order of choice from rows 
for each new term, is represented by writing ab....in\w 
natural order ?i\ times and affixing the /^ suffixes each time in 
a new one of their ?z! permutations. Thus there will be ?/! dif- 
ferent tQvms.^Wxch. are therefore all the terms of A, and the 
sign of any term will be the sign of ( — i)', i being the number 
of inversions among the sufifixes of that term; there being no 
inversions among letters. The algebraic sum of these terms is 
A. This is merely a systematic method of doing what was 
outlined in Art. 25. 

If we select from rozvs in natural order, permuting the 
order of choice from columns, we shall evidentl}^ arrive at the 
same result, but the sign is determined by the number of in- 



i6 INVERSIONS AND DETERMINANTS 

versions among the letters, there being no inversions among 
suffixes. 

28. Let the determinant be represented by I ^1^2 Jn^Xr. 

Then the scheme of development with respect to natural order 
of rows will be identically the same, with regard to syrnbolic con- 
stituents y^^'s^ it was in the development of \a\b-2..,, ///^^ 1^ with 
respect to natural order of columns', but only the symbols of 
the principal diagonal represent the same constituents of the 
determinant in the two cases. Either scheme, however, must 
lead to the development of the determinant represented since 
both accord with the definition of a determinant. 



Represent it by ' a\ bi c^ \c. 



Examples. 

2 — 6 4 

Develop 5 — 3 9 

-7 I -8 

Then letters correspond to columns and suffixes to rows. 
Hence selecting from columns in natural order and permuting 
the orders of choice from rows is indicated by writing a\b2C?, 
a\b-?,c-2, aib\CZy a-ibzcx, azb\c-2, a-ib-icx', the letters being written in 
natural order and the suffixes permuted in all possible orders, 
each permutation corresponding to a term of A. Counting the 
inversions among suffixes in each term, attaching correspond- 
ing signs and substituting the values of the constituents, we 
obtain the 3! terms of A; thus 

2(-i>(«,^2^3) = +(2)(-"3)("8)-(2)(i)(9)~(5)(-6)(-8) 
-r(5)(0(4)+(-7)(-6)(9)-(-7)(-3)(4) 
= 48—18—240+20+378—84 
= 104 
Scholium. To follow tlie symbolic definition (Art. 211 to the 
letter we should write 

A = (-0° (^axb•2C■^) + ( — i)i {a\b^-i) + (-i)^ ia^bxc^) + (— i)^ 
{a-ib:.c\) + (— 1)2 (rtsf^i^s) + {—Yf {a?,biC\)\ but this is unneces- 
sary as the number of inversions in any term is +, or — , ac- 
cording as the number of inversions in that term is even or odd. 

2. Re;)resent A i;i Ex. i. by | axb^c^ \r and develop. Com- 
pare results. Compare Art. 28. 

3. Let the order of choice from columns in Ex. \.\>& b c a 
for every term. Compare results. 

Suggestion, z = number of inversions among botli letters 

and suffixes. 

I 2 I 3 

3 I 4 I 
s. Develop \ .\ ^ Ans. 55. 

I 2 4 I 4 . 



CHAPTER IV 
PROPERTIES OF DETERMINANTS 

29. Theorem. A determi?iant is uimlteredby changi7^gitsroics 

into corresponding colum?is and its columns into corresponding roivs. ^ 2" 

Represent A by ) ^1, ^^2, ;^„ J^ (Art. 27). Change columns 3^^ 

to corresponding rows in the manner of the theorem and repre- ^30 

sent the resulting determinant by Ai. Then Ai = | ^1,^2, ntn\,-. =* S 

But (Art. 28) I ^1, ^2, ntn\c = | ai, h, mn \r. Therefore i °" 10 

A =Ai. O.E.D. 3 S ^ 

* O ^ 

Scholium. Here of course we do not mean tlie arrays arc SS I P 

identical but their determinants are. 2 ^ ^ 

03-- 

30. Theorem. A determina7it is imaltered in absolute value^ '^ W Jo 
hit cha7iged in sign^ by the interchange of any two roivs or any tzvo S "^ *w 
columns. H • M 

m -0 4^ 

'L&tAi=\aib2..hs-'kf'mn\c be the determinant whose col- 30 aj (0 

umns corresponding to h, k, are to be interchanged. Then — ? ^ 

/S.2=.\ aib^. 'ks. .ht. .m^lc is the resulting determinant. De- >^* 

veloping both with respect to natural order of columns y any term Z ^ CO 

of Ai,. as ( — ly a^ b^,. .hp. .kq. .7nu, will appear in A2 as ( — ly (^ PI — 

a^ bw 'kq. .hp. .muy where the order of suffixes is the same ex- » ^ r+ 

cept that p and q are interchanged. Therefore (Theorem I.) n ? 5* 

the signs of the two terms are different. But this reasoning I D 

applies to any term of Ai. Therefore for every term of Ai ^ [J] a 

there is another in A2 of equal absolute value but of contrary .^ pl r« 

sign. Therefore Ai = — A2. 5 1 t 

Let the student erive the proof for the case in which two — z: * 

^ Z rf 

rows are interchanged. r* 1 O 

31. Theorem. The determinant of a square array, in ichicJi 
tzvo rows or two columns are identical y is equal to zero. 

Any interchange in the manner of the theorem should 
change the sign of A. But the value of A is evidently iden- 



]8 



IX\ ERSIONS AND DETERMINANTS 



tically the same in the two cases. Thus A is unaltered in value 
by changing its sign. Therefore A = o. 

32. Theorem. If all the constituents of one line be 7nultiplied 
by the siune factor, the deter tnviant itself zuill be 7nultiplied by that 
factor. 

Let 1^1 <^j m,\c^^I\. Multiply any line as the >^th row, 

or /th column, by ;//. Then all the constituents of the form 
X:,, or all of the form py^ become mx^ or 7npy respectively. 
But every term of A contains one and only one constituent of 
each of the forms mentioned. Therefore every term of A, and 
consequently A itself, is multiplied by m. in either case. 

Cor. From the above and Art. 31 it is concluded that ?//^'^ 
rozcs or tii'O columns differ only by a constant factor the determina?it 
must va7iish. 

33. The following examples are solved by applying one or 
more of the foregoing theorems, as will be evident upon ex- 
amination of the reductions: 

Exercises. 



I 2 4! 
237; 
3 4 10 ! 



I 


4 \ 


I 


7 = 


I 


10 



1 I I 

2 I I 
X I I 



The last step is to subtract tliree times the first column from 
the third. Botli steps might liave been done in one reduction. 



7 II 


4 


7 II 


2 


3 15 


10 


= 6 13 15 


5 


3 9 


6 


I 3 


I 



3 3- 
805 
O O I 



240 



The last determinant develops easily as there is only one 
tf rm not vanishing. 



Jevei 



1 4 7 

2 5 8 

3 6 Q 



7 8 



Thev reduce to 



I 


4 


7 




I 


I 


I 


— and 


I 


I 


I 





I I I 
4 I I 

7 I I 



PROPERTIES OF DETERMINANTS 



19 



7 14 14 

12 Q 24 \= o, because thetliird column is twice the first. 

8 7 16 I 



4 


I 


I 




2- 


—2 


I 


zz 


2- 


8 


I 





230 
460 

—2—8 I 



= o; since in the last deter- 



minant the 2nd row is twice the first. 



6. 



o 3 I 
9 o I 
3 2 I 



q I 
3 o I 

1 6 I 



9 I 

3 9 I 

1 9 I 

The firjt reduction is made by dividing the ist column by 3 
and multiplying the 2nd by 3. 



7 ~ 

3 

4 


I 
6 - 


5 

4 

-3 


= - 


- 


- 


5 --2 7 

4 I -3 

-3 6 4 


= 


I 
12 

I 


I 

—2 

6 


—3 

7 
4 






I 

14 —2 I 

—5 6 22 



4 I 

5 -2 



-313. 



CHAPTER V 
MINOR DETERMINANTS 

34. When any number of rows and the same number of col- 
umns of a square array are deleted, the remaining square array 
is called a minor determinant. The square array of constituents 
which lie at the junctions of deleted rows and columns is also 
a minor. The two minors thus formed are called complement- 
ary minors. Ay^r^/ minor is formed by striking out one row 
and one column; a seco?id minor by striking out two rows and 
two columns; and so on. The first minor formed by suppress- 
ing the row and column through any particular constituent, as 
Xy is called the minor of that constituent and is represented by 
A^. A constituent and its first minor are complementary 
minors. 

^1 bi a 
are first minors of 





^1 bi 




I'^l C\ 


P^^-^ are 




a-2 biL 


> 


az Oi j, 


1 bz c% 


and 


are res 


pe 


ctively 


Ala, A/:2, A.71 



a-2 b-2 c-2 
az bz cz 



35. It follows from the definition of a minor that the 
natural order of its rows and columns is also natural order wnth 
respect to rows and columns of the original determinant. 
Therefore, the niimber of inversions in any term of the minor refer- 
red to natural order of rows a7id columns of the mi?iory is the same 
as the 7iumber of inversions referred to rows and columns of the 
origifial determinant. 

36. Theorem V. The sum of all the terms of/\ which contain 
a given constituent, k, is ( — ly k Ak, where r is the order of k re- 
ferred to rows a7id columns of l\. 

Since no term of A which contains k^ can contain any other 
constituent from the row or column through k, it follows that 
every term of A which contains k must be the product of 
( — ly k (Theorem III and Art. 35) and some term of A*; con- 



MINOR DETERMINANTS 21 

versely, the product of ( — i)'' k and any term, 7}, of A;^ is ob- 
viously a term of A. Therefore the sum of all the terms of A 
which contain /^ is 2' {—\ykTy^ (— i)^ ^2'/; = (— i)''/^ Aa- 

Cor. A deter?ninant of the nth order May be written in tlieforin 
^ kyKy^ wliere ^1, ^2, . . . .>^«, are all the constituents in any line o^ 
AdndKy^{—iy Aky. 

For ( — i)*" ky /\ky is the sum of all the terms of A containing 
ky\ hence -{ — i)''>^^A;^^is the sum of ^// the terms which contain 
an element from the k line of A. But every term of A must, 
by definition, contain one element from the k line; therefore 
there are no terms of A besides those written. 

37. The factors Ki, K2, Kn, are the cofactors of 

^1, /^2,. kn respectively; they are the minors A^^^, A^2»- • • '^k„f 

with positive or negative sign according as the orders of the 
complementary minors k\, h,. . . .k,„ with respect to rows and 
columns of A, are even or odd. When a determinant is ex- 
pressed in the form IkyKy it is said to be developed with refer- 
ence to the line containing the elements k. 

Scholium. It follows that ( — i)<^A/fe is the cofactor of k, if <t is 
the sum of the orders of the constituents of any term of A^t with 
reference to rows and columns of A. 



Scholium. The cofactor of a constituent, axy wil 
sented by the capital letter corresponding, as Ax. 

Exercises. 



)e repre- 



I. Find the sum of all the terms of 



24—1 


14 3 


17—4 


9 12 


3 71 


13 21 


6 -2 


3 6 



which 



contain 13; also of those which contain 9. 

By the preceding theorem and Cor., Art. 32, we obtain, after 

' 24 —I 13 
17 —4 12 =0 and 
6—26 



expanding the last, 13 

= 15876. 
2. Develop 



24 


—I 3 


-3 

6 


7 21 

—2 6 



\ K 

W 



with respect to the 2nd column. 



\ d \}/ ' \ a K \ ' a K 

Ans. A = — X ! „ I — Ai o ' — n , 



INVERSIONS AND DETERMINANTS 



(li d\ ci ih 

il2 bz C2 d2 

as bz cz dz 

a^ b^ Ci d\ 



= ci Ci-\- C2 C% + rs Cz + Ca Ck 





a<i, bi d% 


' 


a\ b\ d\ 


= Cx 


as bs ds 


— C2 


as bs ds 




«4 bi di 




a^ bi d^ 




ai b\ dx 




a\ b\ d\ 


+ ^3 


«2 bi d2 


-~Ci 


«2 bz d2 




ai: bi di 




as bfi ds 



4. What is the cofactor of 9, Ex. 



3613 
-7 —2 7 —5 

4—3 9—6 
5842 



— 414. 



38. Theorem VI. The sum of all the terms of /\ such that 
each term of the sum contaiiu one term of any given minor, Ai, is 
( — i)'AiA2/ A2 bei?ig the complemejitary miiwr and s the sum of 
the orders of the constituents of any term of /\\ with reference to roivs 
and colum,ns of /\. 

Since no term of A which contains a given term ( — i)'X 
{Ilk. . . .^), of Ai, can contain any other constituent from any 
row or column containing /?, k,. . . ,q^ it follows that every term 
of A which contains {Jik....q) must be the product of 

( — \)'^^{]Lk cj) and some term of A2 (Theorem IV. Art. 

35. Art. 20.); conversely, the product of ( — i)* + *"(/^/C' q) 

and any term T oi A2 is obviously a term of A. Hence the 

sum of all the terms of A which contain {hk q) is 

( — \)^^^ {Jik ^)A2. But this reasoning applies to e\'er}' 

term of Ai; s -\-i corresponding. Therefore the sum of all the 
terms of A which satisfy the conditions of the theorem is 

I{-Y)^^i{hk,.,,q)l\^ = {-\yi\2^{—YY{hk.,,,q) 

=(— ij^A2Ai; 
since s is the same for all terms of Ai. 



Scholium. Since the orders of any two complementary 
minors are necessarily both even 01 both odd, s may refer to 
either Ai or A2 indifferently. The generality of the theorem 
also includes this. 



out a 

/ICC- 



MINOR DETERMINANTS 22, 

Cor. A deter mifianty A, of the nth order, may be zcritten in the 
form ^{—lyXmZr; zvJicre any term of the sum consists of the 
product of the sig7i- factor ( — i)^ a m-inor^ as X my selected fn 
given m rozvs {columns^ of /\y and the complementary minor Zr, 
essarily frotn the remaiidng r rows {colu?n7is) ; s being the order of 
the principal diagonal of either X^ or Zr as explained above ^ and no 
two minors of the form X,n bevig selected from the same combination 
of columns {rows) . 

For, every term of the ^ terms of -( — i)'X,nZ^ obviously 
contains m\r\ terms of A, no two of which are alike 
(Theorem VI). Further; no two terms of -, as ( — iyX,„Z^ 
and ( — ly' X,n' Z,.\ can contain a term common to both, since 
X,„ and A^„/ must differ in respect to the constituents of one 
line at least, and one constituent from that line must appear 
in every term of ( — i)'XmZr and ( — i)'' X„/Z/ . Therefore 
~{ — iyA"„^^isthe sum of 77tlr\-^ =7i„—rJ^^- =?«! terms of A, no 
two of which are alike. But this is all the terms of A. The 
corollary follows. 



3Q. Problem. Develop 



ai b\ c\ (h e\ 

a2 bi C2 di €2 

as h cz ds ^3 in the form 

«4 ^4 c^ d\ e\ 

cio br, Ch dn <?o 
^{—\)\axb'^>)(^Cud^ceiS)\ (yaxby) being a minor from the first two 
N| columns. 

Solution. Selecting the ten minors formed by the associa- 
tion of the first two columns with each of the 5-4-^2! combi- 
nations of the five rows two at a time, we obtain {a\b-i), {a\b){), 
(aibi), (aibo), (a2b:]), (^2^4), {a-ibo), (asb^), (aabs), (a^b:,); where 
[ai bi) =.\aib2 Ic, etc., the suffix c being dropped since it is under- 
stood that letters correspond to columns. Hence, enumerating 
orders, attaching sign factors, multiplying by complementary 
minors and summing, there results A = ( — ifiaib^) (csdAeo) 
-hi— 1)7 (aibs) {Cid^eo) + (—1)8(^1^4) {C2d8e6) + (— 1)» (^1^5) {cidm) 
-f {~if{a2b-i) {cxd^es) + {~i? (^2^4) {cidze,)-{-{—i)^^{a2b6) {cid^e,) 
+ (—1)10(^3^4) {cid2e,)-\-{—iYHa^5) (^^2^4) 
-\-{-iYHaAbo){cid2ez)\ 

the mode of development clearly being to write the 10 combina 
tions of the five suffixes, taken two together, in natural order in 
the places of x,y, in {ax by), for the ten minors from the first 
two columns of A. 

fiin^fiin — i) (« — 2) (m — m + i)\s the meaning of the notation employed. By the 

conditions of the theorem we have n^m+r. 



2A INVERSIONS AND DETERMINANTS 



I ai b\ ci (h 

\ a2 bi C2 di 

^ \ az bz cz d^ 



in the form ( — \y {xiVa) {uvv?)\ 

I a\ b\ CJk di 
{x-iyA) being a minor from the 2nd and 4th rows. We must have 
A = ( — 1)» (W4) [cidz) + (—1)^" («2A) {bidz) + (—1)" {a^d^) (bics) 
+ (—1)" ib'iCi) {aids) + (—1)12 (^2^^) (^^^3) _^ (_i)i3 (^2^^) (^aibs) 

Scholium. To find the order of X2y4 simply add 6 to the 
sum of the orders of the letters in every case. 

41. The development of a determinant of the third order by 
the method of Art. 37, and the development of a determinant 
of the fourth order by Arts. 38, 39 and 40, should be thoroughly 
learned by the student. It is not necessary to introduce the 
sign-factor as the proper signs may be attached directly. 



CHAPTER VI 

SIMULTANEOUS EQUATIONS 

42. Theorem. If the elements of one colufnn of a determinant 
be 7miltiplicd in order by the cofactors of the corresponding elements 
of any other column, tlien the sum of the products will be zero. 

Take a determinant of the 4th order, 



7n\ 


n\ 


r\ 


i-i 


JH'i 


n-2. 


r-2 


S2 


ni-i 


n-i 


rz 


S3 


nu 


n^ 


ri 


Si 



multiply the elements of the second row by the cofactors of 
the fourth and take the sum; 

thus: X=m2 M^-^m N^^r^ R\ +52 54. 

But A=M4J/4+^M+ni?4+^4 54. 

Whence ^may be found by changing m.^ to m,^^ iu to ?Z2, and so 
on, in A. It follows that X= o. 

The reasoning applies to columns as well as to rows and to 
any determinant. 

43. Theorem. If each element of a column of a determinant is 
the sum oftivo quantities, the determinant can be expressed as the sunt 
dftwo dctennijiants of the sajue order. 

ii\ b\^{i\ c\ 
Thus; I a-i bi-h c-2 j = (^i+;5i)^i+(^2 + .^2) A -f Uh+^i-s) Pn 
I a-s b-^-Ti'^'i ^3 1 



oil bi ci 




Oi 


ih Oi 


a-2 b-2 c-2 


"T 


^■2 


,u a 


as bz Cd 




ots 


th c^ 



26 



INVERSIONS AND DETERMINANTS 



44. Theorem. A determinant is unaltered in value by adding 
to all the elements of any row the same tnidtiplcs of the correspond- 
ing elements of another row. 



Thus: 



a\ a-i a% 




b\ b-2 bz 


zr 


C\ C-2 c% 





a\-\-mai-^7ia's a-2 a^ 
bi-rmb-2-\rfib^ b-2 b-s 
c\ -{-mc'2 -^-nc'i €-2 cz 



for the second 



a\ a^ a^ 




b\ b-2 bz 


+ 


ei c-2 cz 





member = 



ast two determinants vanish. 



ma2 a<2 az 




mb-2 b-2 bz 


+ 


mc-2 c-2 Cz 





naz 


a-2 


rr.\ 


nbz b-2 bz 


ncz 


c-2 


Cz 



; and the 



45. Let it be required to solve the three simultaneous equa- 
tions 

a\ x-\-b\y-\-ci z == //^i, 
a2 xA^b-2y^C2 z = nti^ 
ciz x-\-bzy-\-cz s — mz. 

First. To find x. Multiply the first equation through by 
A\, the second by ^^2, the third by Az and add. The coeffi- 
cients of .jr,j, ^ in the resulting equation are respectively 

a\ Ai-{-a2 ^2+ az Az = A, 

^1^1+^-2^2+ bzAz= O, 

ci A\-{- C2 A-2-\- Cz As= o; 



where A is the determinant 



ai 


bi 


Cl 


<l-2 


b-2 


C2 


az 


bz 


Cz 



formed of the arra\' oi 



coefficients taken as they appear in the equations. Therefore 
A.r= mi Ai-\-7m A-2-\-mz Az = {mi b^ cz) and x = ( ;;/i b-2 cz)'^A. 

Then multiplying through by BuBi.Bsj instead of Ai, A2, As, 
the value of j^ is found to be (ai7n2Cz) -^ A. Finally, the \alue 
of z is found in similar manner to be {aib-rmz)^^ /\. 



SIMl'LTANEOUS EQUATIONS 
Examples. 

I. Find the values of w,,r,i', 5-, in 



27 



h\ 10 -h x-\-bzy-\-bA 2 = b 
Cl 'W-\-Cq, x-\-C3y-\-C4 s — c 
d\ u>-\-di x-\-(hy-\-ih z=^d 



(^1 b-i c-i d^) {a\ bo cii d^) "^ A 



(^71 b'2 c■^ d] 
A. • 



2. Solve x-\-2y-\-:iz — 6 
2-.r+3J+4^ = 3 
3 -1^+ J'+? 2+1. 



X — 



Solution. 

{m\ b-i r.s) 



{a\ 1912 r.-?) 



6 


2 


3 


3 


3 


4 


I 


I 


^ 


1 


2 


3 


2 


3 


4 


3 


I 


5 



I 6 3 
234 
3 I 5 



4 


2 


I 





3 


I 





I 


4 


I 


2 


2 


2 


3 


2 


3 


I 


2 



I 6 2 
232 
3 I 2 





I 2 6 




I I 4 




2 3 3 




2 I 


(^1 ^2 Wv) 


3 I I 




3—2 



,13 I 
-^i I 4 
I 2 2 
I I o 
300 



I 

—1 


3 


0: 


i-I 


2 


Oj 


1 3 


I 


I 



—6 



—3 



2 I 

3 —2 I _ u 
-6 " 3' 




Scholium, l^emembering the theorems the reductions are 
very apparent, excepting possibly the second reduction for 
tlie denominator of ;r, where the determinant is found from the 
preceding denominator by adding the 2nd row to the 3d and 
subtracting twice the ist from the sum; then the ist from the 
2nd, the first row remaining unchanged. 

Scholium. This method of solving simultaneous eciuations 
should be employed throughout analvtic geometrN 'n order to 
become familiar with the process. 



28 



INVERSIONS AND DETERMINANTS 



46. Homogeneous Equations. In the case of {n-i) homo- 
geneous equations containing ?i unknowns we cannot deter- 
mine the unknowns, but their ratios may be expressed in the 
manner of the following: 

Let ai x^b\y^c\ £r = o, 

ii-2 x-\-b-iy-^c-i 3 = o, 

be two homogeneous equations in three unknowns. Add be- 
low another equation of the same general form, 

as x-\-hy-\-C3 z = \ 

/?3, hy cs, X, being undetermined, and represent the array of co- 
efficients by A. 



.*. X = 






^1 


C\ 





h 


C-2 


X 


h 


cz 



A 






\Bs 



X G 



A 



'Al 



Bs 



a 



The method is easily extended to any number of equations. 



.U = - 



Example. 

2 w-f3.r— 4J'-5 2 = 0, 
^ 7i> - 'x-\-i2 y — 2 z = o, 
6 w — 7 X — 20 y-\- 2 = 0. 

376, Bi = — 752, Ci = 188, D, = 752. 

w X y 2 



CHAPTER VII 

ELIMINANTS AND DISCRIMINANTS 

47. Elimination. To eliminate the /2^<9 quantities x,y, from 
the three equations 

a\ x-\-h y^ci =0, ( I ) 

^2 x-\-b'2y^.C'2 =0, (2) 

az x^hy^Oi = o. (3) 

Multiply (i), (2), (3), in order by Cu C2, 63, and add The 

^1 di C\ 
sum is c\ C\-\-ci C^-\-c% C% = ^2 b^ c^ = p. 

a^ bz C3 

This expresses the condition that the three equations are 
simultaneous in x, y; that is consistent, or capable of being satis- 
fied by the same set of values oi x,y. 

Similarly the four equations 

ai x^b\y-\-c\ s-\-di = o, 

^ a2 x-\-biy^02 z-\-d2 = o, 

az x-\-b-i, y-\-C2, zA^d^, = o, 

a\ x^b^y-\-c^ z-^d\^=^ o, 

between the three quantities x,y, z, are consistent if 



= o. 



a\ 


bi 


Cl 


dv 


a-i 


b.2 


C'2 


d2 


Oi 


bz 


cz 


d. 


aA 


b. 


C\ 


d. 



In general ?i — i quantities involved in a s\^stem of // sim- 
ultaneous equations may be eliminated by arranging the terms 
of each equation in corresponding order in one member and 
equating to zero the determinant of the array so formed by 
the coefficients. This determinant may be called the eliminant 



30 



INVERSIONS AND DETERMINANTS 



or resultant of the system; the reader is referred, however, to 
the higher theory of equations for the precise definition of the 
resultant/^' 

Examples. 

Eliminate ^ and j' from the three equations 

a\ x-^b\ j'+^i z-^di = o, 
a2 x-\-d2y-\-c2 z-\-ih = o, 
az x-^bzy^cz 2-\-ds = o. 



Ans. 



i7i b\ C\2-\-di 

a-i b'2 C'lZ^di 
as bs czz-{-dz 



48. Sylvester's method of eliminating x from any two 
rational, integral equations in x is illustrated by the following 
examples: 

I. Eliminate x from the equations 

ax^-{-bx-\-c — o and ax^-\-^x-\-y = o. 

We derive . ax^-\-bx^-\-cx =d, 

ax--\-bx -\-c =0, 

ax^-\-^A ^-^rix =0, 

ax'^-\-^x-\-y =0. 

Whence, treating x^, x^ and x as so many unknown quanti- 
ties we obtain 

a b c o 

o a b c 

a ^ y o 

\ o a 13 y 



o; 



an equation free of .r. 

2. axr^-\-bx^-{-cx-\-d — <? and px'^-\-qx-\-?' 

Ans. 



a b c d o 
o a b c d 
P q r o o- 
o p q r o 
o o p q I- 



Scholium. It should be observed that the two given equa- 
tions cannot be simultaneous unless the coefficients are such as 
to cause the eliminant to vanish. 



* See Burnside and Panton on Theory of Equations. Vol. II, p. 69. 



ELIMINANTS AND DISCRIMINANTS 



31 



49. Discriminants. It is shown in the theory of equations 
that \if{x) have equal root%f'{x) must ha\'e at least one root 
in common with f{x). It is also shown that/(^') and/'(.i) 
cannot vanish together unless /(;r) have equal roots. There- 
fore the necessary and sufficient condition that f{x) have equal 
roots is that the resultant of the equations /(;i:) —o^f'i^x) =0, 
vanish. When a resultant is formed in this way with reference 
to a function, /(;ir), and its derivate,/'(;r), it is called the dis- 
criminant'^ of the equation /(;i;) =0. Hence the necessary and 
sufficient condition that/(;i:) have equal roots is that its dis- 
criminant vanish. 

Examples. 

I. Form the discriminant of x^ — 2 a x-{-a^ = o. 

2j{x) = 2x^ — 4 a x-\-2 a^, xf {x) = 2 x^—2 a x\ 
:. ■2f{x)—xf' {x) = —-2a x-\-2 a\ /' {x) ^ 2 x -2 a; 

the ehmhiant of which is the discriminant, 



-2a 2^ 
2 — 2a 



-2a 

-2a 



of x^ — 2 a x-\-a-. Hence we infer that j\x) has etjual roots as 
should have been seen at a glance. 

"i 2. P^orm the discriminant of x'^-\-x^ — 5 x-^^t' 

xf'ix) = 3 x^+2 x*~ s^» «/(-*■) = 3 •='^+3 -^^ - ■ 15 -^+9: 
.-. nf[x) — xf {x) = x^ — 10 x-\-g, /'{x)=2 x'^-\-2 -r — 5 



and the discriminant is the eliminant of 

X^ — 10 x'^-\- 9 X 

x^ — 10 jf+g 
3 x^-{-2 X-— 5 X 

3 x'^-j- 2 x—s 



Ans. 



10 9 
I —10 

2— 5 
3 2 



*TIie reader is ay;ain referred to the theory of equations for t'le exact definitions of 
suUaut and Jiscriiuinant. 



INVERSIONS AND DETERMINANTS 

Scholium. A metliod for finding tlie equal roots oi/ix) is 
to equate the H. C. F. oijl.v) and/'(.r) to zero and solve for the 
equal roots. 



3. The discriminant of ao.r^-\-2,aix'^-\-T,a-2X-\-a:>, 
as a determinant is 



o expressed 



iiQ nil a-z 
o ao 2ai ai 
ai 2^2 ^3 



50. We should not confine ourselves to any one method of 
development but learn to make use of several; thus Ex. 2, p. 23, 
solves about as easily by Art. 41 as by the theorems made use 
of. Sometimes, however, one method possesses advantages 
over another and should be employed. 



FEB 10 1903 



LIBRftRY OF CONGRESS 



003 575 566 4 



